AN ALGEBRAIC CHARACTERIZATION OF MINKOWSKI SPACE By RICHARD K. WHITE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA ■V 2001 ACKNOWLEDGMENTS I thank my advisor Dr. Stephen J. Summers for his guidance in the preparation of this dissertation. I would also like to thank all of my committee members for their support and for serving on my committee. TABLE OF CONTENTS page ACKNOWLEDGMENTS ii ABSTRACT v CHAPTERS 1 INTRODUCTION 1 2 A CONSTRUCTION OF THREE-DIMENSIONAL MINKOWSKI SPACE .... 7 2. 1 Preliminaries 9 2.2 Construction of n 11 2.3 Reflections About Exterior Points 18 2.4 Embedding a Hyperbolic Projective-Metric Plane 22 2.5 Exterior Point Reflections Generate Motions in an Affine Space 23 2.6 Conclusion 24 3 A CONSTRUCTION OF FOUR-DIMENSIONAL MINKOWSKI SPACE 26 3.1 Preliminaries and General Theorems 26 3.1.1 Properties oi M 30 3.1.2 Properties of V 31 3.1.3 Properties of X 32 3.1.4 General Consequences of the Axioms 32 3.1.5 Perpendicular Plane Theorems 33 3.1.6 Parallel Planes 34 3.1.7 Consequences of Axiom 11 and 3.1.6.18 37 3.2 Lines and Planes 38 3.2.1 General Theorems and Definitions 38 3.2.2 Isotropic Lines 41 3.3 A Reduction to Two Dimensions 44 3.4 Consequences of Section 3.3 51 3.5 Construction of the Field 52 3.6 Dilations and the Construction of (3£, V,/C) 58 3.7 Subspaces and Dimensions 66 3.8 Orthogonality 69 3.9 The Polarity 80 3.10 Spacelike Planes and Their Reflections 86 ni page 4 AN EXAMPLE OF THE THREE-DIMENSIONAL MODEL 91 5 CONCLUSION 96 REFERENCES 103 BIOGRAPHICAL SKETCH 106 IV Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN ALGEBRAIC CHARACTERIZATION OF MINKOWSKI SPACE By Richard K. White May 2001 Chairman: Dr. Stephen J. Summers Major Department: Mathematics We give an algebraic characterization of three-dimensional and four- dimensional Minkowski space. We construct both spaces from a set of involution elements and the group it generates. We then identify the elements of the original generating set with spacelike lines and their corresponding reflections in the three- dimensional case and with spacelike planes and their corresponding reflections in the four-dimensional case. Further, we explore the relationship between these characterizations and the condition of geometric modular action in algebraic quantum field theory. CHAPTER 1 INTRODUCTION The research program which this dissertation is a part of started with a paper by Detlev Buchholz and Stephen J. Summers entitled "An Algebraic Character- ization of Vacuum States in Minkowski Space" [9|. In 1975, Bisognano and Wichmann [5] showed that for quantum theories satisfying the Wightman axioms the modular objects associated by Tomita-Takesaki theory to the vacuum state and local algebras generated by field operators with support in wedgelike spacetime regions in Minkowski space have geometrical meaning. Motivated by this work, Buchholz and Summers gave an algebraic characterization of vacuum states on nets of C*-algebras over Minkowski space and reconstructed the spacetime translations with the help of the modular structures associated with such states. Their result suggested that a "condition of geometric modular action" might hold in quantum field theories on a wider class of spacetime manifolds. To explain the abstract version of this condition, first some notation is introduced and the basic set-up is given. Let {Ai}iei be a collection of C*-algebras labeled by the elements of some index set / such that (/,<) is an ordered set and the property of isotony holds. That is, if /i,/2 e / such that /i < h, then A^ c Ai^. Let ^ be a C*-algebra containing {yl,}/^/. It is also required that the assignment / 1-+ A, is an order-preserving bijection. In algebraic quantum field theory, the index set I is usually a collection of open subsets of an appropriate spacetime (M,g). In such a case, the algebra Ai is ^ interpreted as the C*-algebra generated by all the observables measured in a 2 spacetime region /. Hence, to different spacetime regions should correspond different algebras. Given a state co on A, let (7^(o,7t(o,Q) be the corresponding GNS representation and let TZj = 7tco(.A/)", / e I, be the von Neumann algebras generated by the 7T(o(^,), / e I. Assume that the map / >-^ Tlj is an order- preserving bijection and that the GNS vector Q is cyclic and separating for each algebra TZ,, i e /. From Tomita-Takesaki theory, we thus have a collection {J/},g/ of modular involutions and a collection {A/},g/ of modular operators directly derivable from the state and the algebras. Each J,- is an anti-linear involution on Jia such that JjTZiJi = TZi' and J,Q = Q. In addition, the set {J/}/6/ generates a group J" which becomes a topological group in the strong operator topology on B(Ti.io), the set of all bounded operators on Tiio. The modular operators {A,},g/ are positive (unbounded) invertible operators such that AfTZjAf = IZj, j ^ I, / e E, i = T-T and A''Q. = Q. In algebraic quantum field theory the state co models the preparations in the laboratory and the algebras Ai model the observables in the laboratory and are therefore, viewed as idealizations of operationally determined quantities. Since Tomita-Takesaki theory uniquely gives these modular objects corresponding to (TZj,Q.), it thus follows that these modular objects can be viewed as operationally determined. Motivated by the earlier work of Bisognano and Wichmann [5], Buchholz and Summers [9] proposed that physically interesting states could be selected by looking at those states which satisfied the condition of geometric modular action, CGMA. Given the structures indicated above, the pair ({^/},g/,co) satisfies the abstract version of the CGMA if {7^,■},■e/ is left invariant under the adjoint action of the 3 modular conjugations {J/l/g/; that is, if for every i,j in / there is a A: in / such that adJiiUj) = JiUjJi = Uk, where JiUjJi = {JjAJi : A e TZj}. Thus, for each / in /, there is an order-preserving bijection, automorphism, X/ on /, (/,<), such that J,TZjJi = Tlx,(j)- fory e /.The set {x,},-6/ is a set of involutions which generate a group T, which is a subgroup of the group of translations on /. Buchholz, Dreyer, Florig, and Summers [6] have shown that the groups Tarising in this manner satisfy certain structure properties, but for the purposes of this thesis, it is only emphasized that Tis generated by involutions and is hence, a Coxeter group. Thus there are two groups generated by involutions operating on two different levels. 1. The group Tacting on the index set /. 2. The group J7 acting on the set {7^/},e/. To elaborate further the relation between the groups Tand J^, consider the following. Proposition 1.1 [6] The surjective map £, : J^^T given by ^(J,, •■■J,-,„) = X/|---T/^, is a group homomorphism. Its kernel S lies in the center of J and the adjoint action of S leaves each 71 1 fixed. ■ Thus, J" is a central extension of the group Tby S. As an immediate consequence of this proposition, J" provides a projective representation of Twith coefficients in an abelian group Z in the center of J. Thus, the condition of geometric modular action induces a transformation group on the index set / and provides it with a projective representation. With this in mind, the following program was then posed. Given the operational data available from algebraic quantum field theory, can one determine the spacetime symmetries, the dimension of the spacetime, and the spacetime itself? That is to say, given a net of C*-algebras and a state co satisfying the CGMA, can 4 one determine the spacetime symmetries, the dimension of the spacetime, and even the spacetime itself? Part of this has been carried out by Buchholz, Dreyer, Florig, and Summers for Minkowski space and de Sitter space [6]. However, in order to do so, they had to presume the respective spacetime as a topological manifold. But would it not be possible to completely derive the spacetime from the operationally given data without any assumption about dimension or topology? As was pointed out by Dr. Summers, a possibility to do so was opened up in this program in the following manner. As already seen, the CGMA yields an involution generated group complete with a projective representation and there is in the literature a way of deriving spacetimes from such groups going under the name of absolute geometry. In general, absolute geometry refers to a geometry that includes both Euclidean and non-Euclidean geometry as special cases. Thus, one has a system of axioms not yet implying any decision about parallelism. In our case, the axioms are given in terms of a group of motions as an extension of Klein's Erlangen Program. A group of motions is defined as a set Q of involution elements closed under conjugation and the group it generates. In a group of motions the representations of geometric objects and relations depend only on the given multiplication for the group elements, without reference to any additional structure. The system of axioms is formulated in terms of the involutory generators alone, so that geometric concepts like point, line, and incidence no longer are primary but are derived. The necessary means for setting up this representation are provided by the totality of reflections in points, lines, and planes (a subset of the set of motions). Points, lines, and planes are in one-to-one correspondence with the reflections in them so that geometric relations among points, lines, and planes correspond to group-theoretic equations among the reflections. This enables one to be able to 5 formulate geometric theorems about elements of the group of motions and to be able to then prove these theorems by group-theoretic calculation. To summarize, we are to find conditions on an index set / and a corresponding net of C*-algebras {^/}/g/ as well as a state co satisfying the CGMA such that the elements of / can be naturally identified with open sets of Minkowski space and such that the group Tis implemented by the Poincare group on this Minkowski space. Out of the group Twe wish to construct Minkowski space such that T's natural action on Minkowski space is that of the Poincare group. This involves two steps. First we carry out the absolute geometry program for three- and four-dimensional Minkowski space. That is, characterize three- and four-dimensional Minkowski space in terms of a group of motions (^,0). Second, we must determine what additional structure on the ordered set / would yield from Tomita-Takesaki theory algebraic relations among the J,- (and hence, among the x,) which coincide with the algebraic characterization found in step one. The organization of the thesis is as follows: in Chapter 2, the given pair (G,<3) is used to construct a three-dimensional Minkowski space out of the plane at infinity. Then identification of the involutory elements of Q with spacelike lines and their group action in with reflections about spacelike lines is made. In Chapter 3 using the same initial data, (^,0), as was given in Chapter 2 but satisfying different axioms, a four-dimensional Minkowski space is constructed. The approach taken here differs from that taken in Chapter 2. This time the affine space is constructed first and then the hyperplane at infinity is used to obtain the metric. The identification of the elements of Q with spacelike planes and their group action in with reflections about spacelike planes is made. In Chapter 4 a concrete example of the three-dimensional characterization is given. As already mentioned, Bisognano and Wichmann showed that for quantum field theories satisfying the Wightman axioms the modular objects associated by 6 Tomita-Takesaki theory to the vacuum state and local algebras in wedgelike regions in three-dimensional Minkowski space have geometrical interpretation [5]. In particular, the modular conjugations, {J/},e/, act as reflections about spacelike lines. In this chapter it is shown that if one chooses the set of wedgelike regions as the index set /, the group JT" generated by the set {J/}/e/ satisfies the axiom system given in Chapter 2 for the construction of a three-dimensional Minkowski space. In Chapter 5 some concluding remarks about the second step described above are made. It is noted that if one assumes the modular stability condition [6] and the half-sided modular inclusion relations given by Wiesbrock [29], then one does obtain a unitary representation of the 2+1-dimensional Poincare group. 4 CHAPTER 2 A CONSTRUCTION OF THREE-DIMENSIONAL MINKOWSKI SPACE In this chapter we give an absolute geometric, that is, an algebraic, characterization of three-dimensional Minkowski space. This chapter is a version of a preprint by the author entitled "A Group-Theoretic Construction Of Minkowski 3-Space Out Of The Plane At Infinity" [28]. Along with the well-known mathematical motivations [1, 2] there are also physical motivations, as we discussed in Chapter 1. Three-dimensional Minkowski space is an affine space whose plane at infinity is a hyperbolic projective-metric plane [12]. In "Absolute Geometry" [2], Bachmann, Pejas, Wolff, and Bauer (BPWB) took an abstract group generated by an invariant system Q of generators in which each of the generators was involutory, satisfying a set of axioms and constructed a hyperbolic projective-metric plane in which the given group (5 was isomorphic to a subgroup of the group of congruent transformations (motions) of the projective-metric plane. By interpreting the elements of Q as line reflections in a hyperbolic plane, BPWB showed that the hyperbolic projective-metric plane could be generated by these line reflections in such a way that these line reflections form a subgroup of the motions group of the projective-metric plane. Coxeter showed in [13[ that every motion of the hyperbolic plane is generated by a suitable product of orthogonal line reflections, where an orthogonal line reflection is defined as a harmonic homology with center exterior point and axis the given ordinary line and where the center and axis are a pole-polar pair. Here we show that Coxeter's and BPWB's notions of motions coincide in the hyperbolic 8 projective-metric plane and that the motions can be viewed as reflections about exterior points. Next we embed our projective-metric plane into a three-dimensional projective space. By singling out our original plane as the plane at infinity, we obtain an affine space whose plane at infinity is a hyperbolic projective-metric plane, three-dimensional Minkowski space. Finally, we show that the motions of our original plane induce motions in the affine space and, by a suitable identification, we show that any motion in Minkowski space can be generated by reflections about spacelike lines. Thus, to construct a three-dimensional Minkowski space, one can start with a generating set Q of reflections about spacelike lines in the plane at infinity. So Q may be viewed as a set of reflections about exterior points in a hyperbolic projective-metric plane. Out of the plane at infinity, one can obtain a three-dimensional affine space with the Minkowski metric, which is constructed from a group generated by a set of even isometries or rotations. The approach in this chapter differs from the method used by Wolff [30] for two-dimensional Minkowski space and by Klotzek and Ottenburg [19] for four-dimensional Minkowski space. The approach in these papers is to begin by constructing the affine space first. For Wolffs [30] two-dimensional case, the elements of the generating set Q are identified with line reflections in an affine plane. For Klotzek and Ottenburg's [19] four-dimensional case, the elements of the generating set Q are identified with reflections about hyperplanes in an affine space. Thus, in each of these papers, the generating set G is identified with a set of symmetries or odd isometries. A map of affine subspaces is then obtained using the definition of orthogonality given by commuting generators. This map induces a hyperbolic polarity in the hyperplane at infinity, yielding the Minkowski metric. To briefly recap the two approaches described above, note that both approaches, ours and the one given by Klotzek and Ottenburg [19] and by Wolff 9 [30], start with a generating set Q of involution elements. In our approach, one can identify the elements of Q with a set of even isometries (rotations) and use the definition of orthogonality induced by the commutation relations of the generators in the hyperplane at infinity to obtain the polarity and then embed this in an affine space to get Minkowski space. In the approach of Klotzek and Ottenburg [19] and Wolff [30], one can identify the elements of Q with a set of odd isometries (symmetries), construct an affine space first, and then use the definition of orthogonality induced by the commutation relations of the generators in the affine space to obtain a polarity in the hyperplane at infinity to get Minkowski space. 2.1 Preliminaries The starting point for an algebraic characterization of Minkowski space is therefore far from unique. Our particular choice of algebraic characterization, in terms of reflections about spacelike lines in three dimensional Minkowski space, is motivated by physical considerations [6] which we briefly explain in the conclusion. A hyperbolic projective-metric plane is a projective plane in which a hyperbolic polarity is singled out and used to define orthogonality in the plane. A polarity is an involutory projective correlation. A correlation is a one-to-one mapping of the set of points of the projective plane onto the set of lines, and of the set of lines onto the set of points such that incidence is preserved. A projective correlation is a correlation that transforms the points 7 on a line b into the lines y through the corresponding point B' . So, in general, a correlation maps each point A of the plane into a line a of the plane and maps this line into a new point A' . When the correlation is involutory, A' always coincides with A. Thus a polarity relates A to a, and vice versa. A is called the pole of a and a is called the polar of ^. Since this is a projective correlation, the polars of all the points on a form a projectively related pencil of lines through A. 10 The polarity dualizes incidences: if A lies on b, then the polar of A, a, contains the pole of b,B. In this case we say that A and B are conjugate points, and that a and b are conjugate lines, li A and a are incident, then A and a are said to be self-conjugate: A on its own polar and a through its own pole [14]. A hyperbolic polarity is a polarity which admits self-conjugate points and self-conjugate lines. The set of all self-conjugate points is called a conic, which we shall call the absolute. In a projective plane in which the theorem of Pappas and the axiom of Fano hold, the polarity can be used to introduce a metric into the plane. Orthogonality is defined as follows: two lines (or two points or a line and a point) are said to be orthogonal or perpendicular to each other if they are conjugate with respect to the polarity. Congruent transformations of the plane are those coUineations of the plane which preserve the absolute; that is, those coUineations which leave the absolute invariant. In a projective plane with a hyperbolic polarity as absolute, the group of all coUineations in the plane leaving the absolute invariant is called the hyperbolic metric group and the corresponding geometry is called the hyperbolic metric geometry in the plane [31]. The conic or absolute, separates the points of the projective plane into three disjoint classes: ordinary or interior points, points on the absolute, and exterior points. The lines of the projective plane are likewise separated into three disjoint classes. Secant lines, lines that contain interior points, exterior points, and precisely two points on the absolute. Exterior lines, lines that contain only exterior points. And tangent lines, lines that meet the absolute in precisely one point and in which every other point is an exterior point. Definition 2.1.1. Two lines containing ordinary points, two secant lines, are said to be parallel if they have a point of the absolute in common. Remark. The set of all interior points and the set of lines formed by intersecting 11 secant lines with the set of interior points, ordinary lines, is classical hyperbolic plane geometry. 2.2 Construction of IT In this section we list the axioms and main results of BPWB [2] and provide a sketch of some of the arguments they used which are pertinent to this work. For detailed proofs, one is referred to the work of BPWB [2]. Definition 2.2.1. A set of elements of a group is said to be an invariant system if it is mapped into itself (and thus onto itself) by every conjugation by an element of the group. An element a of a group is called an involution if a- = 1©, where 1© is the identity element of the group 0. ■ -' •• - « Basic assumption: A given group is generated by an invariant system Q of involution elements. The elements of Q are denoted by lowercase Latin letters. Those involutory elements of that can be represented as ab, where a,b e Q, are denoted by uppercase Latin letters. If ^,t| e and ^r| is an involution, we denote this by 4|r|. Axioms Axiom 1: For every P and Q there is a g with P, Q \g. Axiom 2: IfP,Q \g,h then P = Q or g = h. Axiom 3: Ifa,b,c \P then abc = d e Q. Axiom 4: Ifa,b,c \g then abc = d e Q. Axiom 5: There exist g,h,j such that g \h butj / g,h,gh. Axiom 6.- There exist elements d,a,b e Q such that d,a,b / P,c for P,c e 0. (There exist lines which have neither a line nor a point in common.) Axiom 7: For each P and for each g there exist at most two elements hj e Q such that P\h,j butg,h / A,c and gj / B,d for anyA,B,c,d e <S>(that is, have neither a point nor a line in common). 12 Axiom 8: One never has P = g- We call the set of axioms just given axiom system A, denoted by as- A. The initial interpretation of the elements of Q is as secant or ordinary lines in a hyperbolic plane for BPWB [2]. In our approach, we view the elements of Q initially as exterior points in a hyperbolic plane. After embedding our hyperbolic projective-metric plane into an affme space, we can identify the elements of Q, our generating set, with spacelike lines and their corresponding reflections in a three-dimensional Minkowski space. By realizing that statements about the geometry of the plane at infinity correspond to statements about the geometry of the whole space where all lines and all planes are considered through a point, we see that the axioms also are statements about spacelike lines, the elements of Q, and timelike lines, the elements P of 0, through any point in three-dimensional Minkowski space. The models of the system of axioms are called groups of motions; that is, a group of motions is a pair (0, Q) consisting of a group and a system Q of generators of the group (5 satisfying the basic assumption and the axioms. To give a precise form to the geometric language used here to describe group-theoretic concepts occurring in the system of axioms, we associate with the group of motions (JS>,<3) the group plane (!S>,Q), described as follows. The elements of Q are called lines of the group plane, and those involutory group elements that can be represented as the product of two elements of Q are called points of the group plane. Two lines g and h of the group plane are said to be perpendicular if g \h. Thus, the points are those elements of the group that can be written as the product of two perpendicular lines. A point P is incident with a line g in the group plane if P \g. Two lines are said to be parallel if they satisfy Axiom 6. Thus, \i P ^ Q, then by Axioml and Axiom 2, the points P and Q in the group plane are joined by a unique line, li P )( g then Axiom 7 says that there are at most two lines through P parallel to g. 13 Lemma 2.2.2.[2| For each a e 0, the mappings Ga '■ g ^ g"^ = CLga and Ga '■ P >—>■ P^ = clPol are one-to-one mappings of the set of lines and the set of points, each onto itself in the group plane. Proof: Let a e 0, and consider the mapping y h- >■ y« s aya of onto itself. It is easily seen that this mapping is bijective. Because Q is an invariant system (a* e Q for every a,b e ^) ^ is mapped onto itself, and if P is a point, so that P - gh with g\h, then P" = g"/?" and g«|/7«, so that P« is also a point. Thus, g ^ g^, p ^ p» are one-to-one mappings of the set of lines and the set of points, each onto itself in the group plane. ■ Definition 2.2.3. A one-to-one mapping a of the set of points and the set of lines each onto itself is called an orthogonal collineation if it preserves incidence and orthogonality. Since the " | " relation is preserved under the above mappings, the above mappings preserve incidence and orthogonality as defined above. Corollary 2. 2. 4. [2] The mappings Cf a '■ g ^-^ g'^ ^nd Ua '■ P '-^ P'^ are orthogonal collineations of the group plane and are called motions of the group plane induced by a. In particular, if a is a line a we have a reflection about the line a in the group plane, and if a is a point A, we have a point reflection about A in the group plane. If to every a e one assigns the motion of the group plane induced by a, one obtains a homomorphism of onto the group of motions of the group plane. Bachmann |1] showed that this homomorphism is in fact an isomorphism so that points and lines in the group plane may be identified with their respective reflections. Thus, is seen to be the group of orthogonal collineations of generated by Q. 14 Definition 2.2.5. Planes that are representable as an isomorphic image, with respect to incidence and orthogonaUty, of the group plane of a group of motions (0, Q), are called metric planes. BPWB showed how one can embed a metric plane into a projective-metric plane by constructing an ideal plane using pencils of lines [2]. We shall now outline how this is done. _. ,^ Definition 2.2.6. Three lines axe said to lie in a pencil \i their product is a line; that is, a,b,c lie in a pencil if ■' / - ► abc = d e Q. (♦) Definition 2.2.7. Given two lines a,b with a ^ b, the set of hues satisfying (*) is called a pencil of lines and is denoted by G(ab), since it depends only on the product ab. Note that the relation (*) is symmetric, that is, it is independent of the order in which the three lines are taken, since cba = {abc)~^ is a line, the invariance of Q implies that cab = {abcY is a line and that every motion of the group plane takes triples of lines lying in a pencil into triples in a pencil. The invariance of Q also shows that (*) holds whenever at least two of the three lines coincide. Using the given axioms, BPWB [2] showed that there are three distinct classes of pencils. U a,b\V then G{ab) = {c : c \V]. In this case, G(ab) is called a pencil of lines with center V and is denoted by G(V). If a,b \c then G(ab) = {d : d \c}. In this case, G(ab) is called a pencil of lines with axis c and is denoted by G(c). By Axiom 6, there exist lines a, b, c which do not have a common point or a common line. Recall that lines of this type are called parallel. Thus, in this case G(ab) = {c : c \\ a,b where a \\ b}, which we denote by Px. Using the above definitions of pencils of lines and the above theorems, BPWP [2] proved that an ideal projective plane, n, is constructed in the following way. An 15 ideal point is any pencil of lines G(ab) of the metric plane. The pencils G(P) correspond in a one-to-one way to the points of the metric plane. An ideal line is a certain set of ideal points. There are three types: 1. A proper ideal line gia), is the set of ideal points that have in common a line a of the metric plane. 2. The set of pencils G{x) with x \P for a fixed point P of the metric plane, which we denote by P. 3. Each set of ideal points that can be transformed by a halfrotation about a fixed point P of the metric plane into a proper ideal line, which we denote by /?». The polarity is defined by the mappings GiQ ^ C and C ^ G(0; Px I— ♦ Poo and pao I— ► Px', Gic) 1— ► gic) and gic) h-» G(c). Bachmann [1] showed that the resulting ideal plane is a hyperbolic projective plane in which the theorem of Pappus and the Fano axiom hold; that is, it is a hyperbolic projective-metric plane. In this model, the ideal points of the form G{P) are the interior points of the hyperbolic projective-metric plane. Thus the points of the metric plane correspond in a one-to-one way with the interior points of the hyperbolic projective-metric plane. The ideal points G{x), for jc e ^ are the exterior points of the hyperbolic projective-metric plane. v - . Theorem 2.2.8. Each x e Q corresponds in a one-to-one way with the exterior points of the hyperbolic projective-metric plane. - - Proof: Because each line d of the metric plane is incident with at least three points and a point is of the form ab with a \ b, then each x 6 ^ is the axis of a pencil. From the uniqueness of perpendiculars each x e Q corresponds in a one-to-one way with the pencils G{x). Hence, each x e Q corresponds in a one-to-one way with the exterior points of the hyperbolic projective-metric plane. ■ 16 Thus, the axioms can be viewed as axioms concerning the interior and exterior points of a hyperbolic projective-metric plane. The ideal points of the form G(ab) where a \\ b are the points on the absolute, that is, the points at infinity in the hyperbolic projective-metric plane. Now consider the ideal lines. A proper ideal line g{d) is a set of ideal points that have in common a line a of the metric plane. Theorem 2.2.9. A proper ideal line g(a) is a secant line of the form g{a) = {P,x,G{bc) : x,P\a and abc e Q where b \\ c). Proof: Every two pencils of lines of the metric plane has at most one line in common. By Axiom 7, each line belongs to at most two pencils of parallels and each line g e Q belongs to precisely two such pencils. Thus, a proper ideal line contains two points on the absolute, interior points, and exterior points; that is, a proper ideal line is a secant line. If we identify the points P with the pencils G{P) and the lines X with the pencils G(x), then a secant line is the set g{c) - {P,x,G{ab) : x,P \c and abc e G where a \\ b}. ■ Corollary 2.2.10. The ideal line which consists of pencils G{x) with x \P for a fixed point P of the metric plane consists of only exterior points; that is, it is an exterior line. Under the identification ofx with G(x) then P = {x e Q : x\P}. The last type of ideal line is a tangent line. It contains only one point, G(ab) = Poo, on the absolute. Denoting this line by poo, then Poo = {Giab)} u {x e g : abx e Q}, where a || b. Recalling that each x e g corresponds to an exterior point in the hyperbolic projective-metric plane, we see that a tangent line consists of one point on the absolute and every other point is an exterior point. Also note that under the above identifications, each secant line g{c) corresponds to a unique "exterior point", c, c € g{c) since one only considers those 17 x,P \c such that jcc 9^ 1© and Pc ^ 1©. Each exterior Une corresponds to a unique interior point P and each tangent line corresponds to a unique point on the absolute. Theorem 2.2.11. The map O given by (i) 0(c) = g(c), 0(g(c)) =c (ii) (D(P) = p, 0(p) = P (Hi) O(poo) = Poo, <I>(Pao) = Poo 75 a polarity. Proof: Let V be the set of all points of H and C the set of all lines of IT. From the remarks above it follows that O is a well-defined one-to-one point-to-line mapping of V onto C and a well-defined one-to-one line-to-point mapping of C onto V. Next we show that O is a correlation and for this it suffices to show that C) preserves incidence. Let g{c) = {P,x,G{ab) : x,P\c and abc e Q where a || 6} be a secant line. Let A,B,d,Pa, € g{c), where Poo = G(eJ) = {x e G : xef e G and e ||/}.Then A,B,d\c and cab e Q. ^:'. . '' , •?■ t ^■'h ^{A) = A = {x:x\A}, ^(B) = B, 0(d) = gid), 0(Poo) = p^ = {Poo} <J {x : ejk e Q} cD(g(c)) =c eAnBngid)np^ and a>(g(c)) e 0(^), 0(5), 0(J), <I)(Poo). Hence, O preserves incidence on a secant line. Now consider an exterior line P = {x : x\P} and let a,b e P. Then a,b\P and it follows that P e gia)ngiby, that is, 0(P) e 0(a) n 0(6) and O preserves incidence on an exterior line. Finally, let /7oo = {G(ab)} u {x : abx e Q where o || b} be a tangent line. Clearly, since OiGiab)) = p^o, then Poo = G(aZj) e /?oo. Now suppose that d € /?oo. Then a^^/ e Q and ^(^ = M^ = {A,x,G(ef) : ^,jc|i/ and Je/ e G where e || /}. - Thus, d 6 G(o6) n G(ey) and Poo 6 gid). This implies that d e poo and 0(/7oo) e ^(d). Hence, O preserves incidence and is a correlation. 18 Note also that from the work above, O transforms the points 7 on a hne b into the Unes 0(Y) through the point 'I>(b). Thus, <1) is a projective correlation. Since O^ = 1©, then O is a polarity. Moreover, since 0(pc») - Paa with Poo e /?<», then O is a hyperbolic polarity. ■ Theorem 2.2.12 The definition of orthogonality given by the polarity agrees with and is induced by the definition of orthogonality in the group plane. Proof: If we define perpendicularity with respect to our polarity then the following are true (we use the notation •O' to denote the phrase "if and only if "): (i) g{c) 1 g{d) <^ 0(^c)) = c e gia) and Oigia)) ^ a e gic) <^ a \c. (ii) gic) IP <r^ (^(g(c)) = c e P <-> c IP. (iii) CIP <^ cD(P) = P e gic) = <D(c) <^ P \c. (iv) C Igix)^^ ^(gix)) =x e g(c) = cD(c) <^ x \c. ■ Instead of interpreting our original generators as ordinary lines in a hyperbolic plane, we now interpret them as exterior points. We can construct a hyperbolic projective-metric plane in which the theorem of Pappus and Fano's axiom hold, which is generated by the exterior points of the hyperbolic projective-metric plane. With the identifications above and the geometric objects above, we show in the next section that the motions of the hyperbolic projective-metric plane above can be generated by reflections about exterior points; that is, any transformation in the hyperbolic plane which leaves the absolute invariant can be generated by a suitable product of reflections about exterior points. 2.3 Reflections About Exterior Points Definition 2.3.1. A collineation is a one-to-one map of the set of points onto the set of points and a one-to-one map of the set of lines onto the set of lines that preserves the incidence relation. Definition 2.3.2. A perspective collineation is a collineation which leaves a line 19 pointwise fixed, its axis, and a point line-wise fixed, its center. Definition 2.3.3. A homology \s a perspective collineation with center a point B and axis a line b where B is not incident with b. Definition 2.3.4. A harmonic homology with, center B and axis b, where B is not incident with 6, is a homology which relates each point A in the plane to its harmonic conjugate with respect to the two points B and {b,[A,B\), where [A,B\ is the line joining A and B and {b,[A,B\) is the point of intersection of b and [^,5]. Definition 2.3.5. A complete quadrangle is a figure consisting of four points (the vertices), no three of which are collinear, and of the six lines joining pairs of these points. If / is one of these lines, called a side, then it lies on two of the vertices, and the line joining the other two vertices is called the opposite side to /. The intersection of two opposite sides is called a diagonal point. Definition 2.3.6. A point D is the harmonic conjugate of a point C with respect to points A and B il A and B are two vertices of a complete quadrangle, C is the diagonal point on the line joining A and B, and D is the point where the line joining the other two diagonal points cuts [^,5]. One denotes this relationship by HiAB,CD). Example 2.3.7. Let A,B, and C be three collinear points. For a quick construction of the harmonic conjugate D of C with respect to A and B let Q,R,S be any points such that [2,/?],[!5,5], and [R,S] pass through ^4, 5, C respectively. Let {P} = [A,S] n [B,R], then {D} = [A,B] n [P,Q] ([11]). Note that if [R,S\ \\ [A,Q then D is the midpoint of A and B. Coxeter [13] showed that any congruent transformation of the hyperbolic plane is a collineation which preserves the absolute and that any such transformation is a product of reflections about ordinary lines in the hyperbolic plane where a line reflection about a line w is a harmonic homology with center M and axis m, where Mand m are a pole-polar pair and M is an exterior point. A point 20 . ■ ' - ■ reflection is defined similarly, a harmonic homology with center M and axis m, where M and m are a pole-polar pair, M is an interior point, and m is an exterior line. Note that in both cases, M and /w are nonincident. In keeping with the notation employed at the end of §2. 2, let b be an exterior point and g{b) its pole. r A Lemma 2.3.8. The map T^ : < Ab and d 1 — ► db -\ A' d 1 — > gid)' Pi Poo 1 — ► Poo y A ^ A" d ^ ^d)" y JS. Poo H coUineation. Proof: This follows from the earlier observation that the motions of the group plane map pencils onto pencils preserving the "|" relation. ■ Lemma 2.3.9. ^/, is a perspective coUineation and, lience, a homology. Proof: Recall that g(6) = {A,x,Pca : x,A \b and where b lies in the pencil Pa,}. For ' any A and :>i: in g(b) we have A^ = A and x* = jc since A,x \b and ii A' ,x' g g(b) then A',x' / b and A^ ^ A, x^ * x, and ^*,x* / b. Thus, A'',x^ € gib). Recall also that Poo = G{cd), where c and d do not have a common perpendicular nor a common point and thus, G{cd) = {f : fed e Q). Now g{b) is a secant line, so that it contains two such distinct points, /•<» and goo, say, on the absolute. Since the motions of the group plane map pencils onto pencils preserving the "I" relation it follows that if c,^ e Poo then c*, J* e Poo and hence, P^ = Poo and Qi = Qoo. Moreover, if ;?oo g g{b) then it follows that R^ ^ gib). Thus, 4^/, leaves ""' ''■ ■' ?■> . gib) pomtwise mvariant. •&■► v . j'*,/ ,, ^ Now let g{d), Q, and Too be a secant line, exterior line, and tangent line, respectively, containing b. For e e gid) we have e \d and e^ \ d^ = d since b | d, thus e* e gid). For A e gid), A^ \d^ = d, so A^ e gid). Similarly, it follows that if Poo e gid) then P* e gid), and that gid)'' = gid). One easily sees that Q'' = Q and ri = /-CO. Thus, ^^h leaves every line through b invariant and ^b is a perspective 21 collineation for each b e Q. U Theorem 2.3.10. ^i, 's a harmonic homology. Proof: Since A^ is again a point in the original group plane and since d'' is again a line in the original group plane and from the observations above, we have, for each b e G, ^f, maps interior points to interior points, exterior points to exterior points, points on the absolute to points on the absolute, secant lines to secant lines, exterior lines to exterior lines, and tangent lines to tangent lines. Moreover, since C^*)* = t, for any ^ e 0, 4^^ is involutory for each b e Q. Now in a projective plane in which the theorem of Pappus holds, the only coUineations which are involutory are harmonic homologies [10], thus ^h is a harmonic homology for each b e Q. M Theorem 2.3.11. Interior point rejections are generated by exterior point re/lections. Proof: A similar argument shows that for each interior point A, 4^^ is a harmonic homology with center A and axis A where A is the polar oi A, A € A, and where 4^^ is defined analogously to 4'/,. Thus, each T^ is a point reflection and since A is the product of two exterior points, we see that point reflections about interior points are generated by reflections about exterior points. ■ Theorem 2.3.12. The rejection of an interior point about a secant line is the same as reflecting the interior point about an exterior point. Moreover, since any motion of the hyperbolic plane is a product of line reflections about secant lines, any motion of the hyperbolic plane is generated by reflections about exterior points. Proof: Consider a line reflection in the hyperbolic plane; that is, the harmonic homology with axis g{b) and center b. Let A be an interior point and g{d) a line through A meeting b. Since b e g(d) then b \d and g{d) is orthogonal to g(b). Let E be the point where g(b) meets g(d). Since E e gib) then E \b and Eb =/for some / e ^. It follows that the reflection of A about g(b) is the same as the reflection of A about E. Since b \d and E\d then bd = C and we have E,C \b,d with b ^ d. Thus, by Axiom 2, E = C = bd. Hence, A'' = A""" = AK ■ 22 Theorem 2.3. 13 Reflections of exterior points about exterior points and about exterior lines are also motions of the projective-metric plane; that is, the ^b 's for b e Q acting on exterior points and exterior lines are motions of the hyperbolic projective-metric plane. Proof: The motions of the projective-metric plane are precisely those collineations which leave the absolute invariant. ■ We also point out that the proof that each ^F^ is an involutory homology also showed that the Fano axiom holds, since in a projective plane in which the Fano axiom does not hold no homology can be an involution [4]. 2.4 Embedding a Hyperbolic Projective- Metric Plane In this section we embed our hyperbolic projective-metric plane into a three dimensional projective space, finally obtaining an affine space whose plane at infinity is isomorphic to our original projective-metric plane. Any projective plane Ft in which the theorem of Pappus holds can be represented as the projective coordinate plane over a field /C. (The theorem of Pappus guarantees the commutivity of /C.) Then by means of considering quadruples of elements of /C, one can define a projective space P3(/C) in which the coordinate plane corresponding to Tl is included. If the Fano axiom holds, then the corresponding coordinate field JC is not of characteristic 2 [4]. By singling out the coordinate plane corresponding to 11 as the plane at infinity, one obtains an affine space whose plane at infinity is a hyperbolic projective-metric plane: that is, three-dimensional Minkowski space. To say that a plane n is a projective coordinate plane over a field /C means that each point of n is a triple of numbers (xo,xi,X2), not all x, = 0, together with all multiples (kxo,hc\,'kx2), for X ;t and X e /C. Similarly, each line of 11 is a triple of numbers [«o,"i,"2], not all w, = 0, together with all multiples [Xuo,Xu\,Xu2], A, 9t 0. In PsC/C), all the quadruples of numbers with the last entry zero correspond fr^^-F-^.g— ,»r»-.' -' 23 to n. One can now obtain an affine space A by defining the points of A to be those of P3(/C) - n; that is, those points whose last entry is nonzero; a Une / of ^ to be a Une /' in P3(/C) - n minus the intersection point of the Une /' with FI; and by defining a point P in A to be incident with a Une I oi A if, and only if, P is incident with the corresponding /'. Planes of A are obtained in a similar way [14]. Thus, each point in n represents the set of all lines in A parallel to a given line, where lines and planes are said to be parallel if their first three coordinates are the same, and each line in FI represents the set of all planes parallel to a given plane. Because parallel objects can be considered to intersect at infinity, we call n the plane at infinity. 2.5 Exterior Point Reflections Generate Motions in an Affine Space In this section we state and prove the main result of this chapter. Coxeter showed that three-dimensional Minkowski space is an affine space whose plane at infinity is a hyperbolic projective- metric plane [11]. He also classified the lines and planes of the affine space according to their sections by the plane at infinity as follows: Line or Plane Section at Infinity Timelike line Interior point Lightlike line Point on the absolute Spacelike line Exterior point Characteristic plane Tangent line Minkowski plane Secant line Spacelike plane Exterior line He also showed that if one starts with an affine space and introduces a hyperbolic polarity in the plane at infinity of the affine space, then the polarity induces a Minkowskian metric on the whole space. Under a hyperbolic polarity, a line and a plane or a plane and a plane are perpendicular if their elements at infinity 24 correspond. Two lines are said to be perpendicular if they intersect and their elements at infinity correspond under the polarity. Theorem 2.5.1. Every motion in a three-dimensional affme space with a hyperbolic polarity deRned on its plane at infinity, is generated by reflections about exterior points. Moreover, because exterior points correspond to spacelike lines, then any motion in three-dimensional Minkowski space is generated by reflections about spacelike lines. •if ,.■ . i. t Proof: Because any motion in three-dimensional Minkowski space can be generated by a suitable product of plane reflections, it suffices to show that reflections about exterior points generate plane reflections. Let a be any Minkowski plane or spacelike plane. Let P be any point in Minkowski space. Let / be the line through P parallel to a. Let ttoo denote the section of a at infinity. Applying the polarity to aoo we get a point goo -L aoo. Let g be a line through P whose section at infinity is g-oo, so that g is a line through P orthogonal to a. because each line in the plane at infinity contains at least 3 points, there exists a line / in a which is orthogonal to g as g=o -L aoo. Now let m be a line through P not in a which intersects /. It follows that the reflection of P about a is the same as reflecting m about / and taking the intersection of the image of m under the reflection with g. By the construction of the affme space and the definition of orthogonality in the affine space it follows that / and m must act as their sections at infinity act. because any point reflection in the hyperbolic projective-metric plane can be generated by reflections about exterior points, we have that the reflection of P about a is generated by reflections of P about spacelike lines. ■ 2.6 Conclusion As already indicated above, the geometric model for the generators of Q which lies behind the choice of algebraic characterization of three-dimensional Minkowski 25 space differs significantly from those of previous absolute geometric characterizations of Minkowski space. The model given here is the set of reflections about spacelike lines, which is not a choice which would be made a priori by other mathematicians. However, this is yet another example of a situation where the initial data are imposed by physical, as opposed to purely mathematical, considerations. In the next chapter starting with the same initial data, but satisfying different axioms, a construction of four-dimensional Minkowski space is given. First the affme space is constructed and then the hyperplane at infinity is used. Also given is an explicit construction of the field, the vector space, and the metric. \ , ,• .,.'.•< ,' wiv^v CHAPTER 3 A CONSTRUCTION OF FOUR-DIMENSIONAL MINKOWSKI SPACE In this chapter a construction of four-dimensional Minkowski space will be given using the same initial data as in Chapter 2 but satisfying different axioms. The actual construction is quite long, so to aid the reader in following, A brief outline of the procedure shall be given here. First some general theorems and the basic definitions will be given in the first two sections. In Sections 3.3 and 3.4 attention is restricted to two dimensions in order to obtain the necessary machinery to construct the field. Once the field has been obtained, then a vector space is constructed and given a definition of orthogonality. It is then shown that the vector definition of orthogonality is induced by and agrees with the initial definition of perpendicularity for the geometry generated by the original set of involutions Q. To obtain a metric vector space from the constructed vector space V, a map 7i is defined on the subspaces of V. The map n is defined to send a subspace to its orthogonal complement. Using the work of [3] (which is given for the convenience of the reader) the Minkowski metric is obtained and hence, Minkowski space. The last section of this chapter identifies the elements of G with spacelike planes in Minkowski space and the motions of the elements of Q with reflections about spacelike planes, as was required in Chapter 1. 3.1 Preliminaries and General Theorems Let there be given a nonempty set G of involution elements and the group (d it generates, where for any a e G and for any ^ e we have a^ = ^"'a^ e G- If the product of two distinct elements ^1,^2 e is an involution then we denote this by 26 27 writing ^i |^2- We note that if ^i |£,2 then qj\^2 fo^ ^^^ ^^^xAl iii ® because . ^t^i = ^-i^i^r'^2^ = r'^1^2^ = r'^2^i4 = ^ki- , Let M = {ap: a | P and a,p e ^} and V = Q vj M. We consider the elements of Q as spacelike planes and the elements of M. as Minkowskian or Lorentzian planes. Thus, V consists of the totality of "non-singular" planes and we denote the elements of "P by a, p,y, We begin by giving the basic assumption and by making some preliminary definitions. All the axioms are then listed for the convenience of the reader. The geometric meaning of the axioms and the symbols used will be made clear in the appropriate sections. The first four sections examine the incidence axioms. The order axioms are reintroduced in Section 3.5, where we construct and order the field to obtain a field isomorphic to the reals. In Section 3.6 we give the motivation behind the particular choice of dilation axioms. Using these axioms we are able to define a scalar multiplication and thereby obtain an affine vector space. The polarity axioms are given again and examined in Section 3.8, where we define orthogonal vectors. Basic assumption: If a e Q and ^ & M then a^ e Q and ^^ e M for every E, in Q. For the following, let a, P e "P with a | p. Definition 3.1. If a,p e Q, then aP 6 A^ by definition and we write alp and we say a is perpendicular to or orthogonal to p. ' - Definition 3.2. Suppose that a e ^, P e M.. ' " ■ ^ (i) If aP e Q, then we write a -L P and we say a is perpendicular top. (ii) If aP g Q, then we write al P and we say a is absolutely perpendicular to p. Definition 3.3. Let X = {aP : alp}. We call the elements of X points and we denote these elements hy A, B,C, Definition 3.4. If a,p e M and aP e M, then we write a 1 P and we say a is perpendicular to p. •'I.- 28 Definition 3.5. The point A and the plane a are called inddent when A\a. For each a e V, set Xa = {A : ^ |a}.se that ^,5|ai,a2 where A ^ B and ai 5^ a2. Suppose that A,B\a.\,a.2; where A ^ B and ai ^ ai- We define the line g containing A and B as g = gAB = [ct],ci2] = {C e X : C|ai,a2}. We say that g is the intersection of ai and aa (-^a, and Xaj)^ g <^ oci,a2 {g a Xa\,Xa2)- The Y>omt C \s incident with, g, C & g, ifC|ai,a2. li A ^ B are two points such that there exist a and P with A,B\a,^ and a ± P then we say that A and B are JoinabJe and we write A,B e gAB - [a, P,ap]. If g- is a line which can be put into the form g = [a, P, aP] where a -L P then we say that g is nonisotropic. If A and B are two distinct points such that there do not exist a, P with A,B\fx,^ and a .L P then we say that A and B are unjoinable. If g is a line which cannot be put into the form g = [a, p, aP] with a ± P then we say that g is isotropic or null. ^-^ ^ ; v' ■ f '■ A til y ' '■ ■ Incidence Axioms u - ■ ' , ■ ' .- Axiom 1. For eacli P,a there exists a unique P e "P such tliat P\^ and aP = Q. Axiom 2. 7/^,5 |a,p,s and C\a,^ ttten C\z. Axiom 3. IfP,Q\a,^ and a 1 ^ tJien P = Q. Axiom 4. //a,p,y e G are distinct and a 1 p 1 y 1 a then apy g g. Axiom 5. //a,p e M anda\^ tJien a^ e M. Axiom 6. For ail A, B- A ^ B, there exists a,p sucli that A,B\a,^ and a ^ p. Axiom 7. If a 1 p then there exists A,B\a.,^ such that A ^ B. Axiom 8. ForaJJA,B,C; ABC = D e X. Axiom 9. // 0|a,p,y,5 with p,y,5 1 a then py6 = e. 29 Axiom 10. If A, B, C are pairwise unjoinable points and A, B \ a then C \ a. Axiom 11. For all a e Q, there exist distinct P,y,6 e Q such that a _L P ± y -L ot Axiom 12. IfA,B\a; a e G, then there exists P e G such that A,B\^ and ^ 1 a. Axiom 13. For each P\ a, a e /A, there are distinct points A, B\a such that P ^ A,B and P is unjoinable with both A and B, but A and B are joinable, andifC\u. is unjoinable with P then C is unjoinable with A or with B or C = A or C - B. Axiom 14. If A and B are joinable and A,B\a., then there exists P ± a such that Axiom 15. //a, p,y are distinct with P,y ± a and A,B\a,^,y; A ^ B, then a = Py and ifA,B\y; y J. a then a = Py or p = y. Order Axioms Axiom F. (Formally Real Axiom) [21] Let 0,E \ a, a e M with O and E unjoinable. LetX,b,r\ e V. IfO \ X,5,ri andX,b,r\ L a then there is ay e V such that 0|y, yla, and E^^^^OE^'^^'^ = E^y^^. Axiom L. (LUB) If A (z IC, A ^ 0, and A is bounded above, then there exists an A in K. such that A > X, for all X in A and ifB > X for all X in A then A < B. Dilation Axioms Axiom T. IfO e t,g, with t ^ g, where t is timelike or t and g are both isotropic, then there exists a unique a e M such that gj (^ Xa- Axiom D. (Desargues) Let g,h,k be any three distinct lines, not necessarily coplanar, which intersect in a point O. Let P,Q e g; R,S e h; and T,U e k. Ifgpj \\ ggu and gRT li gsu then gPR II gQs. Axiom R. LetO e g,/T, P,Q e g, and R,S e h. IfgpR \\ ggs then go,POR II go,QOS- 30 Polarity Axioms Axiom U. (U-^ subspace axiom) Let 0,A,B,T and C be four points with 0,C|y6; A,0\a; 0, 5 1 P; with a 1 y and p 1 6. Then there exist X,s e V such that X Iz; 0,AOB\X; andO,C\^. Axiom SI. Ifg cz Xa, CL e G, h c: Xp, ^ e G, and there exist y, 5 e V such that Xld; yd e gn> h, g a Xy, and h a X?, then there exists e e G such that g,h c .^s- (Ifg and h are two orthogonal spacelike lines then there is a spacelike plane containing them.) Axiom S2. Let g and h be two distinct lines such that P e gnh but there does not exist ^ e V such that g c Xp and h c Xp. Then either there are a,p' e "P such that P = ap', g c Xa, and h a Xn' or for all A e g there exists B e h such that P and APB are unjoinable. This concludes the list of axioms. Note that by Axiom 5, ap i A1 for a _L p, a, P e T' and if a, P e M. are distinct with a|P, then.aP e A^. By Axiom 6, every two points is contained in a line. Notation. Due to the brevity of the theorems and proofs in sections 3.1 through 3.4, we shall follow the usual convention in absolute geometry [1, 2, 19, 30] of simply numbering the results in these sections. 3.1.1. Properties of M . 3.1.1.1. The set M * 0. Proof: By assumption ^ ^^ 0, so let a e ^. By Axiom 11, there exists y e ^ such that a 1 y. Hence, ay e A1 by definition and A1 5^ 0. ■ 3.1.1.2. The elements of M. are involutions. Proof: Let y e A^ . Then we may write y = aP where a, P e ^ and alp. We have yy = apap = aapp = 1©. ■ 3.1.1.3. For every ^ e M. and for every l^ e 0, p^ e M.. 31 Proof: Let P = aia2 with ai,a2 e Q and ai |a2. By our assumptions on Q and because ai |a2 we have for all ^ e 0, afaj e Q and a, Ittj so that a'^a^ e M. m 3.1.1.4. The sets Q and M are disjoint, QnM = 0. Proof: Let Q' = Q\M. Then Q' consists of involution elements, Q' o M = 0, and V=g'^M=Q^M. Let y e ^' and ^ e ©. Now if y" e M. then by 3.1.1.3 above y = (y^)^~ G AI. a contradiction. Thus, Q' is an invariant system of generators and without loss of generality, we may assume that Q n M = 0. ■ 3.1.1.5. Ifa e g, ^ e M andal^, then a(3 g M.^ ''^XiX Proof: If aP = y e A^ then a = py e ^ where p,y e tM and P|y, which contradicts Axiom 5. ■ 3.1.2. Properties of P. For the remainder of this dissertation, let the symbol " <->" denote the phrase "if, and only if. 3.1.2.1. //a,p e V and^ e tJien a 1 P if, and only if, a^ 1 P^. Proof: First suppose that a, p e ^. Then a J. p •f^ a^ _L p^ because a ± P implies that a| P and a| P <^ a^ | p^ If a e Q and P e AI and a 1 P then aP = y e ^; a^,y^ e ^, P^ e M, and a^P^ = y^ e Q implies that a^ 1 p^. Conversely, if a^ 1 p^ then a^P^ = 8 e Q and ap = 5^'' e ^, so that a 1 p. Finally, suppose that a.p e M. Then a^P^ e M and ap e AI <^ a^P^ e A^. ■ 3.1.2.2. If a,^ e V and ^ e (3, then a 1^ <^ a'^ 1^^. Proof: Let a e C; and p e At and ^ e 0. If a ip then a|p so that a^|p^. Thus, a^ 1 p^ or a' 1 p^. If a^ 1 p^ then a^p^ = y e ^. So we have ap = y^"' e g by the invariance of ^, which contradicts a ip. Hence, a^ ip^- Conversely, suppose that a^lp^. because a^ e g and P^ g Af by the invariance of g and M then a^ = y, p^ = 5 imply that y 1 5 and y^"' = ai P = 6^'' by the paragraph above. ■ 32 3.1.2.3. For each 4 e 0, V^ = V. Proof: because V - Q^J M, the result follows from the invariance of Q and M. M 3.1.3. Properties oi X. 3.1.3.1. There exists a point; that is, X ^ 0. Proof: Let a e ^. By Axiom 11, there exists y, P e ^ such that a, p,y are distinct and mutually perpendicular. By Axiom 4, apy - a.b € Q, where 5 = Py e A^ and a|5 as a|p,y. Thus, a ±5 and P = a5 is a point. ■ 3.1.3.2. //0|a,p,y,- a,p,y e a; a 1 p 1 y 1 a; then O = apy. Proof: By the proof of 3.1.3.1 above, P - aPy is a point and 6 = Py e A4 because p ± y with p,y e Q. So we have a J_5, 0|6 because 0|P,y. This yields A,0\a,b with ^ == a6 so that A = Ohy Axiom 3. ■ 3.1.3.3. If A e X and 5 e V, then A ^ 5; that is, a point does not equal a plane. Proof: Let A = a^ with a e ^, P e A^, and a ± p. Suppose that ^ = aP = 5 e P. If b & M then we have a = P6 e ^ with P|5 and p,6 e M, which contradicts Axiom 5. If 5 e Q, then A = a^ & Q and this contradicts the definition of a point. ■ 3.1.3.4. The elements ofX are involutory. Proof: Let A e X, so that we may write ^ = ap with alp. In particular, aP = Pa and AA = aPaP = aapp = 1©. ■ ^ :• . . . , „. .: *- "■'■'.:■' -.^h. •■ 3.1.4. General Consequences of the Axioms '^*.-'. *' 3.1.4.1. ///'|a,p andaL^ then P = ap andifP ^ ap thenP\(x,'^ andaL^. Proof: If P|a,p and aip then aP = ^ for some A & X and we have A,P\<x,^ with alp. Thus, by Axiom Z.A^P. HP = ap, then by 3.1.3.4 a I p. because P i. 7^ by 3.1.3.3 then alp. Also Pa = P and Pp = a imply that the products Pa and Pp are involutory so that P|a,p. ■ 3. 1.4.2. IfP I a then Pa e V; that is, Pa is a plane p and P \ p. 33 Proof: By Axiom 1 there exists P such that P \ p and P 1 a. Thus, P | a, P with a i P so by 3.1.4.1 above, P = ap and P = Pa. ■ 3.1.4.3. For each P e X and each ^ e i3, P^ e X. Proof: By the definition of a point we may write P = a^ with a e M , ^ e Q, and alp. By 3.1.2.2, a^ip^ and P^\a^,^^ so that P^ = a^p^ is a point. ■ 3.1.4.4. //i'|a,p andjLa,^ then a = ^.( Given a point P and a plane ^ there exists a unique plane a such that P \ a and a _L y. ) Proof: This follows immediately from Axiom 1. ■ 3.1.4.5. There do not exist three planes pairwise absolutely perpendicular. Proof: Suppose that a, p, and y are pairwise absolutely perpendicular. Then for f = aP we have P | a, P with a, P Jl y so that a = p, which contradicts a J_ p. ■ 3.1.4.6. IfA,B,C\a then ABC = D|a. Proof: By Axiom 8, ABC is a point D and Da = ABCa = oABC = aD. ■ ^.l.A.7. IfA,B,C\a,^ thenABC\a.^. Remark. From 3.1.4.6 and 3.1.4.7 above the product of three coplanar (and as we shall see, collinear) points yields a point which is coplanar (collinear) with the other three. ; " ' . 3.1.4.8. //ai,a2,a3±a then a laja^ = a4 e V and a^ La. Proof: Let A = aai, B = aa2, and C = aas. Then by Axiom 8, ABC is a point D and D = aaiaa2aa3 = aaiaias. So D|a, by 3.1.4.6. By 3.1.4.2, aia2a3 = Da = a4 e V, D = aa4, and aia2a3 = a4±a. ■ 3.1.5. Perpendicular Plane Theorems 3.1.5.1. If a 1 P; Py = ^ andA\a then a 1 y. fA plane perpendicular to one of two absolutely perpendicular planes, and passing through their point of intersection, is perpendicular to both.) Proof: From our assumptions above it follows that ay = a^A = ^Aa = ya, so a i y 34 or a -L Y ( iiote that a = y implies that a i P because A - ^y and A\a). Suppose that a Xy so that B = ay. Then we have A,B\a,y with a ly, which impUes that ^ = 5 by Axiom 3. But ii A - B then Py = ay and P = a, which contradicts P -L a. Thus, ay is a plane and a ± y. ■ 3.1.5.2. Suppose that a ± P; ^|a,p,y,5; yla; and 5±p. Then 5 J. y. {If two planes are perpendicular, their absolutely perpendicular planes at any point of their intersection are perpendicular.) . ,; Proof: By 3.1.4.1, ^ = ya = 6p and 5y = ap e P as a 1 p. Thus, 5 1 y. ■ 3.1.5.3. IfO = aai =yyi = SS| with a,y,s e M. and a J. y J. s J. a then ay = s. Proof: Because a,y,s e M, then ai,yi,Si e Q. Because a J. y J. s ± a and aai = yyi = £8i, then ya = yiai; sa = Siaj; sy = £iyi imply that yi -L ai -L 8] 1 yi. Because 0|ai,yi,8i, then O = aiyi8i by 3.1.3.2. Because points are involutions by 3.1.3.4, loj = 00 = OaiyiSi = Oa\jiOOe\ - aye and ay = s. ■ 3.1.6. Parallel Planes. We say that two planes a and P are parallel, denoted by a || p, if a = p, or there exists a y such that a, pj-y. 3.1.6.1. Parallelism is an equivalence relation on the set of planes V. Proof: That the relation is reflexive and symmetric is clear. For transitivity suppose that a II P and P || y where a,p, and y are distinct. Then there exists 5,8 e P such that a, pis and P,yls. Let A = a5. B = P5, and C = P8. Then by Axiom 8 we have D = ABC = a55pp8 = as and ai.8 so that a || y. ■ 3.1.6.2 If a,^ld and ^le then ale. 3.1.6.3. IfaJ-j, pl5, andy \\ 6 then a || p. (Two planes absolutely perpendicular to two parallel planes are parallel) Proof: If y = 5 then we have a, pi5 and the result follows. So assume that there is an 8 such that y,6l8. Then by 3.1.6.2 above, ai5, ply, a,pls, and a jj p. ■ 35 3.1.6.4. If a II p, y || 5, and^Lb then aly. (Two planes parallel to two absolutely perpendicular planes are absolutely perpendicular.) Proof: because a || p then a, P±s for some s and because P-L5 then aX5. Similarly, y,6±s' for some s' and 5Xp imply that yip. Hence, a,pX5 and P_Ly yield a_Ly. ■ 3.1.6.5. If a e M{Q) and a \\ p then p e G{M). Proof: Suppose that a e ^ and a || P, so a, p±y for some y. Because a e Q then y e tVI, which implies that p e ^. ■ 3.1.6.6. If a II p thena^ \\ p^ for every % e 0. Proof: Let a,p±y for some y e P. By 3.1.2.2, a^,p^±y^ so that a^ || P^. From 3.1.6.5 and 3.1.6.6 above and the invariance of Q and M, we have that if a || P and a e g{M) then p e g{M) and a^ || p^ with alp' e g{M). ■ 3.1.6.7. If a II p then Xa n .^p = ora = p. Proof: Suppose that a,P±y and P\a,^. Then a = P by 3.1.4.4. ■ 3.1.6.8. If A * B then A j( B; that is, AB ^ BA. Proof: By Axiom 6 there exists a e V such that A,B\a. By 3.1.4.1 and 3.1.4.2 we may write A = aa\ and B = aa2. Suppose that AB = BA. Then we have aia2 = a2ai so that aiXa2 or ai ± aj. But ai || a2 because ai,a2ia so that either ai = aj or .^ai (^ ^02 = 0. If ai 1 a2 then by Axiom 7, Xai^Xp ji^ so ai = a2 and ^ = 5. If ai J_a2 then C = aia2 e 3Cai 1^ 3602 ^^^ again we have that aj = aj. Hence, A ^ B. ■ 3.1.6.9.a. IfA'^ = A, then B = A. h. If A ^ B, then A, B, and A^ are pairwise distinct. ■ 3.1.6.10. IfA\a; 5|p; C|y; and a || p || y then ABC \a^y. Proof: Let Aa = a', 5p = p', Cy = y'. Then a',p',y'la,p,y and ABC = aa'pp'yy' = a'p'y' = 5'5 with a'p'y' = S'lS = apy. ■ Conclusion: A\a; 5 1 P; a || p, imply that 5' | p. 36 3.1.6.11. IfAA' = BB'; A,A'\a; B\^; a \\ p then B' \^. Proof: Because^' - BB' then B' = BAA'; p || a || a. From 3.1.6.10 above, B' = BAA' \^aa = p. ■ 3.1.6.12. For each P and each P there Is a unique a such that P\a and a \\ p. Proof: By Axiom 1 there exists a unique y such that yip. because P|y then Pj = a, aiy and a || p. Now ii P\8 and 6 || p then PI 6, a, so that a = 6 by 3.1.6.7. ■ 3.1.6.12. Leta,^ e V and M e X. Then aip <^ a^'ip and thus, a || a^. Proof: Suppose that aXp. By Axiom 1 there exists unique y,5|M such that aXy and 5lp. By 3.1.6.2 we have p,yia and pl5 which implies that yi5. By 3.1.4.1, M = y6. thus a-^' = a^'^ = a^lp^ - p. Therefore, a'^ip and a^ || a. Conversely, suppose that a-^^lp. As above, there exists unique y,6|jV/such that a^iy and 5ip. It follows that pi5, M = y6, and a^ = a^^ = a'^ip and a = (aS)SipS = p. ■ li A-^^ = B, then we say that M is the midpoint of A and B. Clearly, M is also the midpoint of B and A. 3.1.6.14. IfP'* = P'^ then A ^ B. ( Uniqueness of midpoints.) Proof: From P^ = P'^ we have PAB = ABP and also PAB = PBA because ABP is a point and hence, an involution. Thus, AB = BA which implies that ^ = 5 by 3.1.6.8. ■ 3.1.6.15. IfA,B\a,^ andA^ = B then M\a,^. Proof: By Axiom 6 there exists y,5 such that A,M\y,5- because B = A'^ = MAM then by 3.1.4.7 we have 5|y,5. Thus, ^,5|a,y,5 ; M|y,6; and ^,.e|P,y,5 so by Axiom2, A/|p,a. ■ 3.1.6.16. .IfA,B,C\S; a,p,yl5; A\a; 5|p; C|y; a || p || y thenABC\a^y and apyl6. Proof: From 3.1.6.10 we know that D = ABC\a^y = s. Because 5|a,p,y then e|5 so that 8 1 6 or eld. If £ = 5s then we have D,E\5,£ with Sis which implies that 37 D = E hy Axiom 3. Let A = aa', B = pp', and C = yy'. Because a || P || y then a' II p' II y'. Thus, s5 = Z) = ABC = aPya'p'y' = ea'p'y' which implies that 6 = ot'P'y'. Because A \ 5 then we may write A = aa' = 56' and by 3.1.5.1 we have A\a',a,b; aXa'; and a _L 6 which impUes that a' ± 5. Hence, a'5 = a'a'p'y' = P'y' is a plane, so P' -L y'. By Axiom 7 and 3.1.6.7 we have P' = y' so that a' - 5. But this yields ai6 and a J_ 8, which is a contradiction. Thus, aPy ± 5. ■ 3.1.6.17. ///i,B I a; /I I a': a'±a; 5|p; and^ \\ a' then ^ 1 a. Proof: By 3.1.6.16 above B = AAB\a'a'^ = p and p 1 a. ■ 3.1.6.18. //£|a,£; alp, andel^ then s 1 a. Proof: If E\ P then the result follows from 3.1.6.7. Suppose that £ / P and let M = sp. Then, £ ^ £P and £P | aP = a so that E,E^ \ a. Now E^ = f'^P = £P, so that Mis the midpoint of £ and £P. E^^ = £P'=|a'= so that £,£^|a,a'= and by 3.1.6.15, M\a,a^. In particular, M|a and we have Mja, p,s; a _L P; and pis so that a J. e by 3.1.6.7. ■ 3.1.7. Consequences of Axiom 11 and 3.1.6.18. 3.1.7.1. //^|a; a -L p; and a,^ e Q then there exists ay in Q such that A\y and y 1 P,a. Proof: Let ^ 1 6 with 5lp. Then 6 6 A^ and .4 | a, 6 with alp, pl5 so that a 1 5 by 3.1.6.18. Because 5 e M, 6 1a. and a e Q then s = 5a e Q, A\z, and s 1 a. We claim that sip. Indeed, Pe - P5a = 6pa = 6aP = sP with s,p e ^ so that e 1 p or s = p. But s 7t p because then 6a = p and a = 5p = P, as 6ip. This contradicts 3.1.3.3. ■ ' ; 3.1.7.2. Every point may be written as a product of three mutually perpendicular planes from Q. Proof: Let A be any point. Then by definition A = a^ for some a e Q and some P e TW with aip. By Axiom 11 there exists y e ^ such that y 1 a. By 3.1.7.1 there 38 exists a 6 e ^ such that ^|6 and 5 ± a,y. Again by 3.1.7.1 there exists an e e ^ such that A\z and s ± 5,a. Hence, A \ E,a,5; s,6.a e Q, and e J. 5 1 a J_ s which yields by 3.1.3.2, ^ = 5aE. ■ We note that in this case, aP = ^ = a5£ impUes that P = 6z; that is, if ^ | P and P e A1 then there exists Pi,P2 e Q such that /i|Pi,P2 and P = PiP2- 3.1.7.3. If A I a then there Js a P in Q such that A\ P and P -L a. Proof: This follows directly from the proof of 3.1.7.2, for if a e ^ then we can find 5,E e ^ such that ^ |5,e and 6,e -L a. If a e M. then we can find ai,a2 e Q such that .4|ai,a2 and ai,a2 J- a. That is ii A\a then there exists pi,P2 e Q such that ^|Pi,P2 and Pi Jl P2. Moreover, if a e A^ then we can find Pi,P2 such that a = P1P2 and if a e ^ then we can find P1.P2 e G such that aiPip2. ■ 3.1.7.4. If A I a e A^ and P J. a then there is ay such that A\y andy J. P,a. Proof: Let 5|^ with 5lp. Then 6 J. a by 3.1.6.18 and A\z = a5. Because a 1 P and 5J_P then E|p. If eXP then we would have ^|5,e with 5,sJlp so 5 = £ and a = 1©. Thus, £ 1 p. We note that if p e ^ then 6,£ e M and if p e A^ then £ e ^. ■ 3.2. Lines and Planes 3.2.1. General Theorems and Definitions. 3.2.1.1. ForanyA,B e X and any a, b e g, ifA,B € a,b then A = B or a = b. Hence, by Axiom 6, for all A,B e X. there exists a unique g e Q, such that A, B e g. Proof: Let a = [a,p], b = [y,5], ^,5|a,p,y,5, and suppose that A jt B. Let C e a, so that C|a,p. By Axiom 2 it follows that C|7,5 so C € b and a <^ b. Similarly, b (^ a and a = b. ■ 3.2.1.2. Every line contains at least three points. Proof: By definition, every line g = gAB contains at least two points A and B. Let gAB = [a,p] with A,B\a,^. Then by Axiom 8 and 3.1.4.7, ^*|a,p and A^ e X. A'^ ^A by 3.1.6.9. ■ 39 3.2.1.3. Suppose that ap = y. a. If aa i = PP i then a i -L P i and aipi=Y, 50 yj_ai, Pi. b. If aa\ = PPi = TYi then a ± Pi, ai J. p, yi .L ai,Pi, anda^\ = aiP = y. c. If A I a, P then A\y. d. The line g = [a, p] = [p,y] = [a,y] = [a, P,y] Is a nonisotropic line. Thus, a,^,and y are tJiree mutually perpendicular planes which intersect in a line. Proof: a. From aai = PPi we have y = pa = Pitti. . b. If PPi = yyi and aai = yyi then yi = yPPi = aPi and yi = yaai = Pai. c. Because /4 1 a, p and ap = y then Ay = Aa^ = a^A = yA and A \ y. d. By Axiom 7 there exist points A and B such that A,B\a,^ and by 3.2. 1. 3. c. above, A,B\y. Thus, A,B & [a,P],[a,y],[p,y] and [a,p] = [a,y] = [p,y] = [a,p,y] is a nonisotropic hne. ■ 3.2.1.4. If A and B are collinear then A^ and B^ are collinear for any t, e 0. Proof: This follows from the fact that v4,5| a, p ^^^,5^ |a^,p'. ■ For each ^ € 0, we define a^ = {C^ : C e a}. By 3.2.1.4 above, a^ is a line for every ^ in © and if « = [a,p] then a^ = [a^,P^]. Definition of parallel lines and planes. We say the line a is parallel to the plane a, denoted by a \\ a, if there exists a P such that a c 3Cp and P || a; that is, ^ = [P'Y] ^iid a, P 1 6 for some 5. We say that two lines a and b are parallel, a \\ b, if there exists a,P,y,5 such that a = [a,y], b = [p,6], where a || P and y || 5. 3.2.1.5. IfAA' = BB' ^ 1©; A,A'\o.,o!\ a ^ a'; 5.B'|p,p'; p ^ p', then g^^' \\ ggg>. Proof: Let fi|p* with p* || a (3.1.6.12). Then fi'|p' by 3.1.6.11. Let B\^\, with Pt II ai. Then 5'ip; by 3.1.6.11 and 5,5'|p,Pi,p',p^ which implies that [P.Pi] = [P%P|] = gBB' by 3.2.1.1. Hence, we have g^^i = [a,ai], ggg' = [p*,Pt] with a II p* and ai || Pi; thus, g^^< \\ ggg>. ■ 40 3.2.1.6. Two lines, a - [a,ai] and b - [p,Pi] are parallel precisely when there exist A, A' e a and B,B' e b such that AA' = BB' * l^. Proof: Suppose that a \\ b. Then, a || P and ai || Pj. Let A, A' e a and B e b, so that B' = AA'B\aa^, and aiaipi = p,pi, by 3.1.6.10. Thus, B,B' e b with AA' = BB' . li AA' = BB' = 1& then A = A' . ■ 3.2.1.7. If a II b and b \\ c then a \\ c. Proof: Let a = [a,a']. From the proof of 3.2.1.5 above we may write b - [P,p'] with a II P and a' || p' because a \\ b, and c = [7,7'], where y || p and y' || P' because b II c. By 3.1.6.1, a || y and a' || y' so that a || c. ■ 3.2.1.8. For each line a and point A there is a unique line b such that A e b and Proof: Let B,C e ahe distinct and if A e a we can choose B,C ^ A because every hne contains at least three points by 3.2.1.2. Then ABC = D by Axiom 10 and BC = AD * ig, so gAD II a by 3.2.1.6. Now suppose that A & c and c \\ a. By 3.2.1.7 above, c \\ gAD, so there exist W,X e c and 7,2 e gAD such that WX = YZ ^ \& by 3.2.1.6. By 3.1.4.7, ^' = ^fFX e c and ^1 = /iFZ e ^.4Z). Thus, lg3 ^AA' = WX = YZ = AAx impUes that^' = Ax. Rence, A,A' e gAD,c; A ^ A' , because WX ^ 1© and gAD = c by 3.2.1.1. ■ 3.2.1.9. IfA,B e g, A^ B, andC e h then g \\ h if and only if ABC eh. Proof: If ^5C = H e h, then 1© ^ AB = HC and g \\ h. Conversely, suppose that g II h and put D = ABC. Then because A i^ B, 1© ^ AB = DC and C e gDC, with gDc \\ g- Because ^ || A and C e h, then by 3.2.1.8, gDc = h and D e h. M 3.2.1.10. If a II b then a = boranh = 0. Proof: Suppose a \\ b and a r^ b ^ 0. Let A e a,b and C e a with C ^ A, B e b, with A it B. Then, a = gAc, b = gAB, and therefore, D = ACB e a,b. ■".'T- '■■ •-,."=y'-"i " 41 U D = A, then ACB = A, C = B, and a = b. If D ^^ A, then it follows that « = gAD = gAC == ^BD = gAB = b. M Classification of nonisotropic lines. Following the terminology of physics we make the following definitions. Let o be a nonisotropic line. If there are elements a,p,y e M such that a = Py and a = [a, P,y], then we say that a is timelike. If there are elements a, P e ^ such a 9^ P and a = [a, p,a6], then we say that a is spacelike. Remark. Let A,B\a e Q with A ^ B. Then by Axiom 14, there is a P such that ^,5| P and P ± a. Thus, a = [a, P] is a spacelike line by definition, A,B e a, and every pair of points in a spacelike plane is joinable with a spacelike line. ■■,.>»' ^'■" t 3.2.2. Isotropic Lines. f ; „ ' }*.v 3.2.2.1. If A ^ B; A,B\a,^ with P ± a, a e M; and A and B are unjoinable with P; P\a,y; andjLa,^, then A'' = B. Proof: First, /i^|a^,P^ = a, p. U A^ = A then A\y and A is unjoinable with P. Similarly, 5^|a,p and B^ ^ B. Suppose that A'' and P are joinable. Then there are 5 and s such that P,A''\8,e and 5 1s. But then P,A = /'T,(.4Y)r|6V,8T and 6^ 1 s^, which implies that P and A are joinable. Thus, A"^ and P are unjoinable, so by Axiom 13, A"^ is unjoinable with A or A'^ is unjoinable with B or A"^ = A or A'' = B. Hence, Ay = B. m 3.2.2.2. Suppose that A ^ B; A,B\a,^ with a L ^.^ a e M; A and B are unjoinable withP\a; P|y, yip. then A^ = B. Proof: First observe that because P\a,y; alp; and yip, then a 1 y by 3.1.6.18. Then A^la^, p^ = a,p and A^ is joinable with B. A^ ^ A because A and P are unjoinable as in 3.2.2.1 above. ^^ joinable with P implies that A = (A^y is joinable with py = P. Hence, by Axiom 13, Ay ^ B. M 3.2.2.3. If A and B are unjoinable then X e g^B precisely when X = A orX^BorX is unjoinable with both A and B. 42 Proof: By Axiom 6, gAB = [a. P] for some a and p in V. Let X e g^B and suppose that X ^ A.B. If X is joinable with A then there exists y,6 such that 7X8 and A,X\^,b. By Axiom 2, 5|y,5 and A and B are joinable. Hence, Xis unjoinable with both A and fi. ■ Remark. If Xis unjoinable with J and B; A,B\a.,^; then by Axiom 10, ^|a,P and X e gAB- 3.2.2.4. If gAB is isotropic; C,D e gAB', and C ^ D. then C and D are unjoinable. Proof: If C,D|a,p with a ± P then A,B\a,\i by Axiom 2; that is, gAB = gCD- ^ 3.2.2.5. IfA,B,C\a are pairwise joinable and distinct, then each P\a is joinable with at least one ofA,B, and C. Proof: If a e ^ then the result follows from Axiom 12, so assume that a e A^. By Axiom 13 there exist D,E\a such that A,B, and C are all unjoinable with P. Then by Axiom 13, at least two of A,B, and C must lie on one oi gpo and gpE. By 3.2.2.4 above, this implies that two of A,B, and C are unjoinable, which contradicts our assumption. ■ 3.2.2.6. If gAB is isotropic then g\g is isotropic for all '% e <S> . Proof: If.4^,5^|a,p with alp then ^,5|a^"',p^"' and a^"' 1 p^"'. ■ 3.2.2.7. IfP\a e M then there are at most two isotropic lines in Xa through P. Proof: If gAP,gBP,gcP (= '^"a are three distinct isotropic lines through P, then P is unjoinable with A,B, and C. The points A,B,C must be pairwise distinct and are joinable by Axiom 13, in contradiction to 3.2.2.5. ■ 3.2.2.8. By Axiom 13 and 3. 2. 2. 7, for each P\ a, a e M, there are precisely two isotropic lines in Xa through P. . , . .. 3.2.2.9. Let gAB,gAC <= -^a, a & M, be two isotropic lines. a. If A I p. with pla, then g^^g = gAc- b. IfA\y, withyla, then g^g =gAC- 43 Proof: Because g,4B * gAC by assumption, then B is joinable with C by Axiom 13. The results follow ftom 3.2.2.1 and 3.2.2.2. ■ 3.2.2.10. IfA,B,C\a, a e M; gAB '*■ Sac are isotropic then ggc is nonisotropic and there is a^ with ^\A such that P 1 a and C = B^. Proof: Because g^B "*■ gAC^ then gBC is nonisotropic. By Axiom 14 there exists a y, y|5,C, such that y ± a. Let ^ | P, P 1 y (Axiom 1). Because ^ | a, P; a ± y; and ply, then by 3.1.6.18, a -L p. If B^=B then B\ P and g^g is nonisotropic. Similarly, CP ^ C and by 3.2.2.1, ^P - C. ■ 3.2.2.11. IfA,B,C\a. a e M. are pairwise unjoinable then there exist ^,y ± a with A\^,y such thatC = B^y. Proof: First observe that by 3.2.2.3 and 3.2.2.4, gAB = gBC = gAC C= -^a is isotropic. By 3.1.7.3 and 3.1.7.4, there exist ^\A such that P -L a. Because B and C are unjoinable with A, then 5,C / p and B^ ^ B. By Axiom 1 and 3.1.6.18, there is a 6|P such that 5 1 P and 5 1 a. Thus, 5,5P|5,5P = 6,a and B and fiP are joinable. Suppose that ^P and C are unjoinable. By 3.2.2.3, either gg^(^ = g4c or A is joinable with 5P. But iiA,B^\£, s 1 a, then ^ = A^,B\e^ 1 a^ = a imply that /4 and B are joinable. If g^p^^ = g^c = gAB then ^P is unjoinable with B. Hence, ^P and C are joinable and by 3.2.2.10, there is a y with y\A such that y 1 a and C = fiP^ . ■ 3.2.2.12. LetP\a,^,y with p,y 1 a, a e M. If gAB <= -"^a is isotropic then g% c Xa is isotropic and gAB II g%. Proof: By 3.2.2.8, there are precisely two isotropic lines in Xa through P, say gpc and gPQ. By 3.2.2.9, g^pc = gpg and g% = gpc- Now PAB = Z) is a point by Axiom 8, so AB = PD and D is unjoinable with P as ^ is unjoinable with B. (Suppose that P,D\a' with a' 1 a. Let 5|X, with X \\ a'. Then because B\a, X 1 a by 3.1.6.17. But then by 3.1.6.16, A = PDB\a'a'X = ?. and ^ is joinable with B.) Because 44 A,B,P\a then D = PAB\a. Hence, D e ^/-c or D e gpg, and g^a || gpc or , By II By m By m Py n h gAB II g/^e- thus g% II g^^ = gPQ II g^fi or g^Jj II gpc - g/>C II gAB- ■ 3.3. A Reduction to Two Dimensions In this section we restrict our attention to two dimensions. In this way we are able to use the work of Wolff [30] to construct our field. Let ^ be any element in 0; then the map a^ : X >-^ X given by C7f(^) s ^£,~' is bijective and maps lines onto lines and planes onto planes by 3.2.1.4,5,6 and 3.1.2.3. Hence, it is a colHneation of (0,^,X). In the following, for any element 4 e 0, the collineation induced by it is denoted by ajc. Let a e P be fixed throughout this section. We wish to define a set Ca of maps on Xa which can be viewed as line reflections in a given plane Xa- We then show that each element of Ca is involutory and that Ca forms an invariant system of generators within the group Ga it generates. Finally we will show that {Ca,Ga.) satisfy Wolff's axioms [30] for a two dimensional Minkowski space; that is; the Lorentz plane, ior a e M.. Let Ca - {g <^ Xa '■ g is nonisotropic}. By Axiom 14, each g e Ca may be written uniquely in the form g = [a, P,y] where a = Py. Let A e Xa and g = [a,p,y] e £«• because A \ a and a = py, then A = ^" = A^'' and A^ - A^. Hence, we define the map Cg : Xa >—>■ Xa by ag(A) = A^ = A^ = A^ , for A e Xa, and ■ ag : £a ^ Ca by (5g{h) = h^ = {ag{A) : A e h). Note that by 3.2.1.4, if , h = [a,5,s], then Ggih) = [a,6P,sP] = [a,6^si']. 3.3.1. For each g = [a,p,y] e £«, <yg '■ Xa '—>■ Xa is bijective. Proof: liA,B\a and ag(A) = OgiB) then A'' = B^ and A = B, so Cg is injective. If y4]a then ^^ ] a^ = a because a ± y and Cg(A^) = {A''y = A, so Og is onto. ■ 3.3.2. Forg = [a,p,y] e £«, cig : Ca '->■ Ca Js bijective. Proof: Let h = [a,s,r|] e £«• Because a 1 y and s 1 r| <-> e^' 1 r|^ by 3.1.2.1 then 45 a = a^ = s^ri^. It follows that Cgih) - [a,8^,Ti^] e £«• If ^|e,ti then ^^|s^,ti^ so Cg(A) e C!g{h) for all A eh and Og is a well-defined collineation. If / = [a,A.,n] e Ca and Cg{h) - dgil) then for every A in h we have CTg(.4) ^Aye Ggil) = [a,X\^y]. This implies that A = (/i''')''' e [a,A,,[i] = /;that is, h - I. Hence, Cg : Ca '-' Ca is injective. Because s ± r\ <r^ z'^ 1 r\'' then it follows that h = [a,£,r|] e Ca <r^ Og(h) = [a,s^TiT] e £«• : -'•<, Hence, CTg : £a '— *■ Ca is surjective. ■ 3.3.3. Each a g, for g e Ca, is involutory. Proof: Let g = [a,p,y]. Then for ^ | a we have OgCSg(A) = ag(A'') ^ (A^y = A. ■ Let g = [a,5,y], h = [a,s,r|] e Ca- We say that g is perpendicular to or orthogonal to h, denoted g -L h, if one of y and 5 is absolutely perpendicular to one of s and r| and gn h ^ 0. 3.3.4. Forg and h as above, ifyle then 5±r|, 5 1s. andy 1 s. Moreover, A/= yc = 5r| e g,h. Proof: Let M = ye. Because y,s 1 a then Af- = (ys)" = y"s" = ys = M and M\a. So we have M\a,e which implies that M|aE = r| and M e h. Also, M\a,y so M\ay = 5 and Meg. Also from M = ys it follows that Ms = y = 6a = 5ti£ so that M = 8r\ and 5iTi. Applying 3.1.6.18 to M|y,Ti with y 1 5, Til5, and M|6,s with Sly, sly, we obtain y 1 r| and 5 1 e. ■ 3.3.5. Ifa,b,g e Ca then a 1 b <^ a^ 1 b^. Proof: By 3.1.2.1 and 3.1.2.2 we have a c Xy- yl5; b ^ Xs <r^ a" cz X^; y^15^; b'^ c xl, for all ^ e (5. The result follows. ■ 46 Remark. Because A\s <r^ A^\z^ for all ^ e 0, A e X, and s e 'P, it follows that for each g e Ca. CTg maps: (i) The set Xa onto itself. (ii) The set £a onto itself. (iii) Collinear points in Xa onto collinear points in .^a- (iv) orthogonal lines in Xa onto orthogonal lines in Xa- That is, Gg is an orthogonal collineation of Xa- Let Ca = {cTg : ^ e £«} and *Pa = {a/> : P e Xa}- 3.3.6. The sets Ca and Xa are nonempty. Hence, Ca ^ and'^a "*■ 0. Proof: If a e Al then we may write a = a'5 where a', 5 e Q and a' ± 6 by definition of M.. By Axiom 7, there exist A,B such that A ^ B and A,B\a',b. Thus, A,B\a'6 = a and Xa ^ 0- Moreover, g^B - [a, a', 5] e Ca and Ca "^ 0- li ct. e Q then by 3.1.3.1, there exist A\a and by 3.1.7.3, there is a p in ^ such that A\P and P ± a. Thus, A e Xa and g = [a,p,aP] e Ca- ■ Note that from 3.3.3, Ca consists of involutory elements and by 3.1.3.4, ^a consists of involutory elements. 3.3.7. IfP e g,h e Ca and g 1 h then Op = OgOh = ^h^g- Proof: Let g = [a, 7,5] and h = [a,8,ri], and without loss of generality, assume that yls, so that 5ir|. By 3.3.4, P = ye = dr\ e g,h and by 3.2.1.1, {P} = gnh. Then for A e Xa, ogOhiA) = Gg(A^) = A^y = Gp(A) = Ay^ = GhiAy) = uhGgiA). m 3.3.8. For every A In Xa and for every h e Ca, there is a unique g e Ca such that A e g and g 1 h. Proof: Let h = [a,s,Ti] e £« and A e Xa- By Axiom 1, there exist ay\A such that yls. Thus, A\a,y with a 1 s and yls so a 1 y by 3.1.6.18. because a 1 y then ay = 8 and A\a,y implies ^ 1 5 so that A e g = [a,y,5] e Ca- because g c Xy, yls. 47 A e g = [a, 7,5] e Cn- Because g c Xy, yJ-S, and h c Xe, then g -L h. By 3.3.4 above, tiX5 and {B} = {ys} = {t|5} = gnh. Now suppose that I a Ca such that A e I and I 1. h. Now we may uniquely write / = [a,a',P] in jEq where a = a'p. because I ± h then without loss of generality, we may assume that a'_L£ and P-Lr|. because y-Le and 81.r\ it follows that a' II y and P jj 5. By the definition of parallel lines in Section 3.2 we have / || g. because /i e /ngthen by 3.2.1.10, I = g. ■ 3.3.9. (i) The fixed points of a g e Ca are the points Jn Xa incident with g. (ii) The fixed lines ofog e Ca consist ofg and all lines in Ca which are orthogonal to g. Proof: Let g = [a,y,6]. li A - C!g{A) - A^ for A \ a, then A \ y. because A'' = A^, ^A \ a then A^ =AandA\ 5. Thus, A e g and 3.3.9. (i) follows. Let h - [a,s,ri] € £« and suppose that Ogih) = h; g ^ h. Then there is an A e h such that A € g. By 3.3.8, there is a k - [a,co,0] e £« such that A e k and k ± g. Because A e h, OgiA) = A'' e h and because A e k, k 1 g then ^^ I (0^,0^ = (0,9 and ^^ e k. Because A ^ g then A )( y (for if y4|y then A\a implies that ^ jay = 5 which implies that A e g) and A * A^. Thus we have A,A'^ e h,k, so that h = khy 3.2.LL Hence, h 1 g. M 3.3.10. (i) The only fixed point of a point reHection, Gp, is the point P. (ii) The fixed lines ofop are the lines incident with P. Proof: If P' = P, then this implies that APA = A, or AP = l<s, or A = P. Thus, 3.3. 10. (i) holds. Let x = [p,u] be any line (not necessarily in Xa or nonisotropic) and A e X. Then P-^ e x implies that P-^ \ p,u, so that Fj p'^,u-< = p,u, and P e x. M Let Ga be the group acting on Xa generated by Ca- By 3.3.5, every element of Ga is an orthogonal collineation of .^a- Let ct e ^a- By a transformation with a, we mean the adjoint action of or on ^a : on e Ga ^ of = ctctiCT"' e ^a.Note that every such transformation is an inner automorphism of the group. 48 3.3.11. Let h = [a,E,ri], g = [a, 7,6] e £«, then Gg* = (s^^(g). Proof: Let A \ a and put B = OgiA) = A'^. Then Thus, (Tg*(a/j(^)) = ffCT/,(g)(cf;i(^)) for all A e .^a- Because a>, is injective, then CTg'' = (To^{g) for every A in ^a- Similarly, C!g''(a) = ao^,(g)(o), for every a in Ca- Hence, dg'' - Oai,(g)- Analogously, we can obtain ct^'' = a^^^^p), for every P in Xa- B Because Ca generates Ga and every element of Qa is an orthogonal coUineation of {Xa,JO.a) then we obtain the following. 3.3.12. For every (J e Qa and for every Og e Ca, we have ct| e Ca', that is, Ca /5 a/2 invariant system of Q a- B Consider the two mappings: g e Ca ^-^ ^g ^ Ca and P e .^a *— >• <7/' e ^a- The first one is from the set of nonisotropic lines in £« onto the set of line reflections, and the second is from the set of points in .^a onto the set of point reflections in Xa- These mappings are injective, because reflections in two distinct points have distinct sets of fixed points by 3.3.10, and similarly for lines and line reflections. Thus, it follows from 3.3.11, 3.3.12, and the preceding remark that the next result obtains. , - - 3.3.13. //a e Ga and P,Q e Xa then aiP) = Q <^ af = cjq. Proof: Because the result follows. ■ 3.3.14. Iffy e Qa and g,h e £« then G(g) = h <r^ a^ = o/j. ■ 3.3.15. IfP e Xa andg e £« then P e g <^ apCg is involutory. Proof: Suppose that P & g= [a,y,6] e £„■ Then for .4 | a, cpdgiA) = (yp(A'') = Af^ = A'^ = OgGpiA). Thus, {cpOgy = l.i-„. Now assume that 49 apug is involutory. Then P = a gO pg gcy p(P) = P^^^ = P^'^. Because yPy ^ P^ e Xa by 3.1.4.3, then by 3.1.6.9, P = P^. Because P'^ = P^ then P\y,d, and hence, Peg. m 3.3.16. Ifg,h e Ca, then g 1 h <r^ CgOh is involutory. Proof: Nowg ± /? if and only if ag(h) = h. For g ^ h, by 3.3.9, Cgih) = h li and only if a^^ = a/,. For Gg -^ u/, by 3.3. 14, a'^/ = a/, if and only if (agG/,)^ = Ixa and GgGh ^ l.^a- ■ 3.3.17. TJie point reflections ^a are Z^ize involutory products of two line reflections from Ca- Proof: Let ap e ^a where P e Xa- Then we may write P = ySs where a = y5 and y,6,s e ^ by 3.1.7.3. Let g = [a,y,5]. Because a = y5, then g e Ca- By 3.3.8, there is an / in £« such that Pel and I -i- g. By 3.3.7, Gp = CT/CTg = <7gCJ/. ■ We now show that if a e A1, then the pair (Ca,Ga), acting as maps on (j£a,'Ca,a), satisfies Wolff's axiom system [30] for his construction of two-dimensional Minkowski space. We give Wolffs axiom system below. Basic assumption: A given group &, and its generating set Q of involution elements, form invariant system iG,<3) The elements of Q will be denoted by lowercase Latin letters. Those involutory elements of (3 that can be represented as the product of two elements in Q, ab with a I b, will be denoted by uppercase Latin letters. Axiom 1. For each P and for each g, there is an I with P,g\l. Axiom 2. IfP,Q\g,l then P = Q or g = I. Axiom 3. IfP\a,b,c then abc e G. Axiom 4. Ifg\a,b,c then abc e Q. Axiom 5. There exist Q,g,h such that g\h but Q / g,h,gh. Axiom 6. There exist A, B; A ^ B, such that A, B ^ g for any g e Q. (There exist unjoinable points.) 50 Axiom 7. For each P and A,B,C such that A,B,C\g, there Is a v e Q such that P,A \ v orP.B\v orP,C\v. Geometric meaning of the axioms. The elements denoted by small Latin letters (elements of Q) are called lines and those elements denoted by large Latin letters (thus the element ab with a\b), points. We say the point A and the line b are incident li A\b; the lines a and b are perpendicular {orthogonal ), \{ a\b. Further we say two points A and B are joinable when there is a line g such that A,B\g. Replacing £« with Ca and Xa with ^a, the pair {Ca,Qa) satisfies the basic assumption by 3.3.3, 3.3.6, and 3.3.12. It follows from 3.3.15, 3.3.16, and 3.3.17, that our definition of points, our incidence relation, and our definition of orthogonality agree with those of Wolff. Hence, we may identify £« with Ca, Ca with Q, and Qa with 0. Verification of Wolff's axioms. Axiom 1 follows from 3.3.8 and Axiom 2 fi:om 3.2.1.1. For Axiom 3, let a = [a,a',s], b = [a,p,p'], c = [a,Y,y'] € Q. Then a',p,y 1 a, and by our Axiom 9, a'Py = 6 1 a and d = [a,5,a5] e £„• For yi|a, (SaObCciA) = ^tP«' = ^«'Pt =A^ = Gd(A). Hence, CaObOc = cj^ e Ca. If we then identify "|" with "e", we get Wolffs Axiom 3. Axiom 4. Ifa,b,c\g, then abc e Q. , ,. -i* „ . . '\''' Proof: Let g = [a,XXl a = [a,a',5], b = [a,p,p'], and c = [a, 7,7'], with a',p,7 1 X. Then by 3.1.4.8, a'py = z L X. Let A e a, B & b, and C e c, then ^|a',a; 5|a,p; and C|a,7. By 3.1.6.10 /15C = D|a'p7 = e and .45C = D|a by 3.1.4.6. Thus, Z)|E,a; z LX; a L X, and s 1 a. So, d = [a,s,as] e £«• And if X|a, then UaOhCciX) = ^P«' = J^ = a^(A) and UaObGc = a^ e Ca M Axiom 5: There exist Q,g,h such that g\h but Q / g,h,gh. 51 Proof: Because a e M, we may write a - Py, where P,y e Q and p J- y. By 3.2.1.2 every line contains at least three distinct points. Let g - [a,p,y] e Ca and let fi e g. By 3.3.8, there is an h in Ca such that B in h and h ± g. By 3.2.1.2, there is an A in h such that A ^ B. By 3.3.8, there is an / e Ca such that A e I and I L h. By 3.2.1.2, there exists Q e I such that Q ^ A. Now a Q e h, then /i,(^ e l,h implies that Q = A or I = h, so Q H h. Suppose that Q e g. Let Q e m. m J- I, so that cq = diGm- Because ag = GgO/,, 04 - CT/,ct/, and BAQ - E e Xa, then <JE = '^B<^A<^Q - ^g^h^h^l^t^m = ^g^m and g -L W. Because Q e g,m and g 1 w then cr^ = <jgCSm = O/Gm and Og = Cj ov g = /. This implies A,B e Um. Thus A = B or I = m, a contradiction. ■ Axiom 6: There existA,B;A ^ B, such that A, B / g for any g e Q. Proof: This follows from our Axiom 13. ■ Axiom 7: For each P and each A,B,C such thatA,B,C\g, there is av inQ such that P,A\v. orP,B\v, orP,C\v. Proof: By our Axiom 13, there exist two isotropic lines in .^a through P, say gpg and gPK. If no such v in £« satisfies the above, then two of the three points must lie on one of the isotropic lines by our Axiom 13. But this implies that gpg = g or gPR = g', that is, g is isotropic; a contradiction. ■ 3.4. Consequences of Section 3.3 3.4.1. For every A and B, there exist a e M such thatA,B\a. {Every line lies in some Minkowskian plane.) Proof: By Axiom 6, there exist a & V such that A,B\a. II a e M, the result follows. So suppose that a e ^. By Axiom 12, there exist p e ^ such that A,B\^ and p ± a. Then A,B\a^ = y, and y e >( by the definition of M. ■ 52 3.4.2. For every A and B, there exists M such that A^ = B; that Js, every two points has a midpoint and by 3.1.6.14, the midpoint is unique. Proof: By 3.4.1 above, there exists a g A^ such that A,B\a. From Section 3.3, there exists M\a such that A^ = B. ■ 3.4.3. IfAA' = BB' then a and A' are joinable precisely when B and B' are. Proof: Suppose that A and A' are joinable and let A, A' |a,a' with a -L a'. By 3.4.2, there exists an M such that A''^ = B. Then B = A^\a^,a"^ and from 3.1.2.1 and 3.1.6.13 it follows that a^ 1 a'^, a || a-^', and a' || a'^. By 3.1.6.11, AA' = BB'; A,A'\a,a'; B\a^,a'^; a \\ a^'; and a' || a'^. Thus, B' I a^,a'^ and, B and B' are joinable. ■ 3.4.4. If a. 1 a'; a || P; a' || p'; and [a,a'] \\ [p,p'], tlien p 1 p'. Proof: Because [a,a'] |i [P,p'], then there exist A,A'; A,A' \a,a' and there exist B,B'; 5,5' I pp', such that AA' = BB' . Then for A^ = B, as in the proof of 3.4.3, [P'P'] =^BB' - [a^^a'^] II [a,a']. That is, 5|a'^^a'^p,p'; with p,a^ || a, and p',a'^ II a'.Thus, p = a^ and P' = a'^' by 3.1.6.12. Hence, p ± p' because 3.5. Construction of the Field The basic construction. For the construction of the field we will follow the path of Lingenberg |20|. Throughout this section let a e M and (9|a be fixed. Define the sets: Oa = {g e Ca : O e g) and Pa(0) = {(Tga/, : g,h e Oa}. Proposition 3.5.1 The set T>aiO), acting on the points ofXa, is an abelian group. Proof: By 3.1.7.2 we may write O = ap = yrip with a = yx\; y,Ti,p e g-, and y,y\, and P mutually orthogonal. Thus, g = [a,y,Ti] e £„: O e g, and Oa ^ 0. Because each 53 Gg e Ca is involutory then Ixa ^ T^aiQ). Now let Oa^'^b^^c^d ^ T^a{0) where a = [a,a',aa'], b = [a,p',ap'], c = [a,y',ay'], and d - [a,5,a5] are nonisotropic. Now 5y'p' = E ± a with 0|s by Axiom 9 and/= [a, 8, as] e Oa- Thus, for A\a, Hence, GaOhOcOd = GoGj e Va(0). Because each a/ is involutory for / e Ca, CT^' = cj/ and {GaGhY^ = GhGa e Va{0). From Axiom 9 and the calculation above, the product of any three of 8,7', P', and a' is an involution. Thus, ^8y'p'a' ^ ^P'y'Sa' ^ ^P'a'Sy' ^^^ GaGbGcGd = GcG^GaGi,. That is, VaiO) is abelian. Clearly, VaiO) is associative so that T>a{0) is indeed an abelian group. ■ Lemma 3.5.2. Let g be an isotropic line in Xa with O e g. Then for every GiGi, e VaiO), G/Ghig) = g. Proof: Let / = [a,p,aP] and h = [a,y,ay] be Hues in £« with O e l,h. By 3.2.3.12, if 0| P,y with p,y 1 a then gPv || g. But OP^ = O so that CT/,a/(0) = O^^ - O. Thus, gPY =ghy 3.2. L 10. Hence, ci;,cj/((9) = g, for all (j/,a/ e VaiO). ■ Lemma 3.5.3. Z/e^g be an isotropic line in Xa through O. Then for every A, E e g, E ^ O, A * O, there is a unique g/Gi, e VaiO) such that g/G/XE) = A. Proof: By 3.2.3.11, there exist y,5 1 a with 0|y,5 such that £T^ = A. Take / = [a,y,ay] and h = [a,6,a5]. Note that li E = A then E^^ = E implies that F' = E^ or E' = E^, which implies that / = h [30]. To show uniqueness, suppose that GaGh,GkGi e VaiO), where / = [a,5,a6], k = [a,s,a8], g = [a,y,ay], and h = [a,p,ap] are lines in XaiO) and E ^ E''" = £*. Then £''«*' = E and ^P^'^S = E. By Axiom 9, pys = X with 0\X, XI a. Thus, m = [a,X,a'k] e OalGkGaGh = Gm, and E = E"''. Let £| a',5' with a' Lx, 5'i6, and put M = a'X and N - 5'5. Now £|a',a,5' with a IX, a 1 ?t, 6' 1 5, and a ± 5, so that a', 5' 1 a by 3.1.6.18. It follows that M^ = a'«;t« = a'X = Mand ^« = 5'«6« = tV . So that M\a,X ■ N\a,b; and m - goA/ and / = goN. Then because E'"' = E, we have E^^ = E; E^ = E^ and 54 £A/ = ^a'x ^ £8'8 ^ £A' and M - ^ by 3.1.6.14. So w = ^o.w = ^o.v = / and a/ = a,„. Therefore Oa^h = ^A<^/- ^ Let us denote this unique map by 6^ . So for all A e go£, /I ^ O, (i) 6.,(0)-0 (ii) 5^(£) = /J (iii) 5.4 e Pa(0). Translations. For every pair J, 5 of distinct points we can define a translation Tab '■ ^ '— *■ -^ given by T^siA) = AAB = 5. We now restrict our attention to a set of translations defined on .^a and we note that [iA,B,C\a, then D = ABC\a by 3.1.4.6. Thus, Tab '• X.a '-^ Xa is a well-defined map for all A,B\a. Let /C = goE be an isotropic line in Xa and define Ta = {Tqa '■ A e K,}. Theorem 3.5.4. The set Ta is an abelian group. Proof: Let Tqa^Tob e Ta. Because C = BOA = AOB e IC, for Xla, we calculate (ToA ° 7-05)^ = ToAiXOB) = XOBOA = XOC = Toc{X). Thus, Tqc = Tqa ° Tqb e Ta. Also, Too{X) = XOO = X. Hence, Too e Ta is the identity l^a on Xa- To find r5|^, we compute (ToA ° 7^0.4'^ )W = ^O^^O^ = XOOAOOA = X = XO^O^^ = {Tq^o o ToA)iX). Hence, 7^^ = T^ao, A^ = OAO e IC, and 7^^ e Ta. Clearly, ToA ° (Tob ° Tqc) = {Tqa ° Tqb) ° Toe for A, B,C e /C. Therefore the action of Ta is associative and Tx is a group. Because ABC is a point and hence, an involution for all points A,B,C, then for A,B e IC and X\a, iToA ° Tob)(X) - XOBOA = XOAOB = {Tqb ° ToA)iX). Therefore, Ta is abelian. ■ Lemma 3.5.5. For all Tqa e Tx, ro.4(/C) = K.. Proof: Because Ov45 e IC for A,B e IC then ro.4(X;) c X; for all Toa e Ta. Now let 55 C e IC and A e IC. Then OCA = D e )C so that C = ODA and ToAiD) = C. Hence, each ToA ^ %. maps ^ onto IC. M Lemma 3.5.6. Let Tqa e Ta and g c Xa- Then ToAig) = g if and only if g is parallel to /C. Proof: Let gnr == g be a line in Xa such that ToAigHp) = gHF- Then ToAiH) = HOA e gHF and g///r || K by 3.2. L9. Conversely, suppose that gHF II ^• Then again by 3.2. L9 it follows that for B e gHF, Toa(B) = BOA e gHF and ToA(g) = ^- ■ Lemma 3.5.7. IfA,B e h, h a Xa^ and h \\ /C, tJien there exists Tqc ^ %i such that Toc(A) = B. Proof: By 3.2.L9 we have C = OAB e /C and it follows that B = AOC = TociA). ■ Lemma 3.5.8. For each A e IC, there is a unique Tqa e %l such that Toa(0) = A. Proof: Clearly, ToAiO) = OOA = A. So suppose that TociO) = A. Then ooc = c = A. m We denote this unique translation mapping of O into A hy Ta- Lemma 3.5.9. If a e VaiO) and Ta e Ta, then oTao-^ = T^^y Proof: Let ct = OgCy, where g = [a,P,aP], h == [ay, ay] e Oa- Then for any Xja, By 3.2.3.8 there exist precisely two isotropic lines in Xa through O : K, = goE and ^' = gOF- We define multiplication and addition on the points of IC so that the points of IC form a field. For A,B e IC, define: A + B ^ (Tb o TaXO) A • B = {dB° 6 .()(£), where A,B * O and £ e /C is the multiplicative identity. A'0=0'A = 0. Theorem 3.5.10. For every A, B e IC, Ta+b = Ta ° Tb. 56 Proof: For X\a, Ta+bW = Taob(X) = XOAOB = XOBOA = {Ta o TbKX). ■ Theorem 3.5.11. For all A,B ^ O in /C, 6^.5 = 8,4 o 6^. Proof: Let 5^ = CTad^' and 6a = cr^a^'. Then A-B = 658^ (£) = £«'«*'*. Put Gc = a^'(7„CT^', then, ct^cTc e Va{0) and a/,cjc(£) = £^* = £"'^*'* = A - B. Hence, by 3.5.1.3, dA-B - ^b^c = ^b^b'^a^a " ^a^^h'^b^a' = f^a^o'<^6^6' = ^A^B- ■ Hence, (/C,+) is a group isomorphic to Ta and (/C\{0},-) is a group isomorphic to VaiO). It remains to show that the distributive laws hold. Theorem 3.5.12. Let A,B, C e IC, then (A + B) - C = A - C + B • C. Proof: If C = O, then (A+B)'C = 0=0-0+0'O^A'C + B'C. li C 7^ O, then we compute (A+B)-C = 5ciA + B) = 5cTa+b(0) = 5cTa+b^cH0) - ScTaTb^cHO) = dcTA^c^dcTBd = nc{A)nc(B)iO) = TA.cTB.ciO) = TA.c+B.ciO) = A'C + B-C. Because multiplication is commutative, C'iA+B) = iA+B)-C^A'C + B'C = C-A + C-B. Hence, (/C,+,.) is a field. ■ Ordering the field and obtaining R. To order the field K. we make use of the following [21]. Let F be a field and ^,,... ,/!„ e F. If /f,,... ,^„ ^ implies that X^=i Aj ^ 0, then F is called formally real f Theorem 3.5. 13. (Artin-Schreier) Every formally real field can be ordered. ■ To make the field /C formally real the following axiom is posited. Axiom F. (formally real axiom). LetO,E\a, a e M with O and E unjoinable. Let X,5,r| e V. IfO\X,b,r\ andX,b,r[ 1 a then there is ay e V such that 0\y, y 1 a, and E'^^^^OE^'^'^'^ = £^t^y. Theorem 3.5.14. // Axiom F holds on K, then /Cis formally real. Proof: Let / = [a,?.,a?.] e Oa and let dgcs,, e Va{0). Then from Proposition 3.5.1, 57 CgGijG, = a I for some I e Oa and CTga/, = ct/ct,. Clearly, if /' e Oa with a^'G/ = ct/ct; then we have a/- = a/ and /' = /. Hence, for all agCJ/, e VaiP), there is a unique / e Oa, such that cjgCT/, = CT/CT,. So if O ^ ^4 e /C, then we may uniquely write b^ in the form GaCT,; that is, for every O ^ A e /C, there is a unique a, a = [a,a',aa'] e Oa such that A = aaO,(E) = E'" = £^«'. Suppose that O^ A = £^"'. Then ^2 = £>.a'Xa' ^ ^ implies that £ - QaXaX ^ Q^ a contradiction. Hence, if O ?t ^i e /C then A~ ^ O. Now suppose that A,B e }C and ^,5 ?t O. Let o = [a,a',aa'], b = [a,p,aP] e Oa such that A = £^" and B = £^P. Then by Axiom F there exists a y g P such that 0|y 1 a and A^+B^ = £^a'^a'o£^P^P = E^^^y. (3.5.1) Since 0|y 1 a then c = [a,y,ay] e Oa,Oca, e Pa(0), and C = GcGtiE) = E" = E^y e /C. So equation (3.5. 1) reads A- + B^ = C^. If C^ = f^v^^Y = O, then E = O^'^^^ = O. Since E ^ O, it follows that if ^i,...,/l„ e K, are all nonzero, then Y.l=\Al * O and /C is formally real. ■ To finally obtain a field isomorphic to the real numbers, M, we add the least upper bound property to our axiom system. '■ Axiom L. 7/0 ^ AciK. and A is bounded above, then there exists an A e }C such that A>X, for all X e A, and ifB e KL with B > X for all X e A then A < B. The only ordered field up to isomorphism with the least upper bound property is the real number field, R. Theorem 3.5.15. The field IC constructed above, along with Axiom F and Axiom L, is isomorphic to the real number field, R. 58 In the next section an affine vector space is constructed from products of pairs of points. The scalar multiplication is obtained by adapting and extending the definition of multiplication of elements of K. 3.6. Dilations and the Construction of {X.V.K) The additive group V and dilations. First we construct a vector space V over the field /C. Let V = {OX : X e X}. First note that the product, AB, of any two points A,B ^ X, is in V because AB = 0(OAB) = OOAB. We view the elements OX € V as directed line segments with initial point O and terminal point X on the line gox- We define an addition on V by setting OX+OY= OXOY. The product of three points is a point, so XOY = Z e X, OXOY = OZ e V. Theorem 3.6.1. (V,+) is an abelian group. Proof: Let X, Y,Z e X be distinct. Then, 0X+ OY = OXOY = OYOX = 0Y+ OX, and addition is abelian. The zero vector. is 1© since 1& - OO e V and 1© + OX = leOX = OX = OX+ 1©. To complete the proof we calculate 0X+ 0X0 ^ OXOXO = OXOOXO = OXXO ^00= 1©, so -OX^ oxo. (OX+ OY) + OZ= {OXOY)OZ = OX{OYOZ) = 0X+ {OY + OZ). Hence, (V ,+) is an abelian group. ■ We still need to define a scalar multiplication of K on V. To do this note that in an affine space the group of dilations with fixed point C is isomorphic to the multiplicative group of the field. Noting this, we geometrically construct such a group of mappings and use these mappings to define our scalar multiplication. A dilation of X is a mapping b : X ^ X which is bijective and which maps every line of X onto a parallel line. [23, p.37] Theorem 3.6.2. [23, p.42| A dilation 5 is completely determined by the images of two points. 59 Proof: Let 5 : J£ i-» JC be a dilation and assume that 5(A) = ^ and 5(7) = T' of two points X,Y e X are known. We must show that the image of any Z e X is known. Suppose that Z g gxr- Then clearly Z ^ X and Z ^ Y and we consider the two lines g^ and gYz- Observe that Z e gxz ^ grz- If these lines had a point in common besides Z, they would be equal and then gxz = grz = gXY- But this is impossible because Z g gxr- Therefore, {Z} = gxz f^ grz- Since 5 is bijective, from set-theoretic reasons alone, d(gxz ^ gvz) = ^igxz) ^ ^(grz); or equivalently, {5(Z)} = d(gxz) f^ ^(gvz)- In other words, the lines 5(gxz) and digyz) have precisely one point in common, namely, the point 5(Z) for which we are looking. The line 8(gxz) is completely known because it is the unique line which passes through X' and is parallel to gxz- Similarly, the line 5(gYz) is the unique line which passes through the point Y' and is parallel to gYz- because 5(Z) is the unique point of intersection of ^(gxz) and 8(gyz) (3.2.1.1), the point 6(Z) is completely determined by X* and 7*. Conversely, assume that Z g gxy- If Z is Jf or Y, we are given 5(Z), so assume that Z 1^ X and Z t^ Y. By 3.4.1, there is an a e A1 such that X, Y\ a and there exists a P I a such that P ^ gxy. Then Z € gxp and from the previous paragraph, d(P) is known. Hence, using the line gxp instead of the line gxy, we conclude from the earlier proof that 5(Z) is known. ■ To define a scalar multiplication, we fix a timelike line / and use it to geometrically define dilations. To aid in the construction, the following facts are used to add the appropriate axioms. In a three-dimensional or four-dimensional Minkowski space, if / is any timelike line through a point O and g is any other line through O, then there is a unique Lorentz plane containing the two lines. Two distinct isotropic lines intersecting in a point in Minkowski space determine a unique Lorentz plane. Desargue's axiom, D, holds in any affine space of dimension d > 3. 60 Axiom T. IfO e t,g where t is timelike or t and g are both isotropic then is a unique (X e. M. such that g,t a Xa- Axiom D. Let g\, gj, and gj, be any three distinct lines, not necessarily coplanar, which intersect in a point O. Let P\,Q\ e g\; P2,Q2 e g2; and Pt„Qj, e ^3. // SP\Pi II gQxQi ^ndgp^p^ II gQ^Q^ thengp^p^ \\ gg^g^. Axiom R. LetO e gugi; P\,Q\ e gi; and Pj.Qi e g2. Ifgp^p, \\ gg^g^ then gO,P\OP2 = go.Q^OQi- Axiom T refers to the first statements. Axiom T is used to put an isomorphic copy of the field on every isotropic hue through O. A scalar multiplication is then defined in a manner similar to the definition of multiplication for the field elements. Axiom D is Desargue's axiom, the "dilation" axiom. Axiom D ensures well-defined dilations with the standard properies of such maps. Axiom R is used to distribute a scalar over the sum of two vectors. The dilations are constructed next. Let IC c Xa, a & M.By 3.1.7.2, there exist ai,a2 e Q such that 0|ai,a2 and a = aia2. Because Oja, then Oa = p e ^ and O = a^ - aia2P with o.\,a2 X p by 3.1.6.18. Let y = aiP e M. Then ya = Paiaia2 = Pa2 e M, so that t = [a,y,ay] c Xa is timelike and O e / as 0|a,ai,a2,P implies that 0|a,y,ay. So let t = [a,y,6] e Oa be any timelike fine through O and )C' c Xa the other isotropic line through O. Then for each At e t, there is a unique A e K, and there is a unique B e K.' such that At = AOB. Because Ai e t, then B ^ A'. (From this point on denote a line reflection GgiX) by X^. Thus, X^'"' means arCf,Og(X) = agCi,c!r(X).). For each A, e /, there is a unique A e K. such that At = AOA'. Similarly, for each A in IC, there is a unique At in / such that At = AOA'. Thus, there is a one-to-one correspondence between the points of IC, the field, and the points of /. Fix Et - EOE' as the unit point on /. For each O ^ A e IC, we use t to construct a dilation 6"^ of X in the following way. Let X e X. 61 If X /, then by Axiom T, there is a unique r\ e M. such that goxJ <^ -^ti- From Sections 3.5 and 3.6, j£ti is an affme plane and our definition of parallel lines in ^n is equivalent to the affine definition. So, there is a unique line h c: Xri such that A, e h and h || gxE,- Because goxnt = {O} ^0, hnt = {A,} * 0, and h \\ gE,x, then hngox = {B} * 0. In this case, set E^iX) = B. li X & t and X ^ O, then because t,K, a Xa with a e M, there exists a unique g (z Xa such that A & g and g \\ gxE- Because gxE r\t = {E] t- then i ^n/= {fi} ^ 0. Set 6^(A) = fi. ^, .SV If X = O, set hA{X) = O. Note that 5^(A) e ^oj^ by construction. It is clear that 5^ : ^ H^. X is one-to-one. We find it useful to make some observations. Let O ^ ^ e /C and recall that £ e /C is the multiplicative unit point; E ' A = A, io^ A ^ K. VvX At = AOAK IjQmm&^.Q.Z. For the map b A defined above, the following are true: 1 ?>a{E)=A. 2 ?>A{E,) = At. 3 ^A is onto. Proof: I. E, = EOE' and A, = AOA' imply that EEi = OE' and AAt = OA'. If EE, = OE' = le then O = E' and O = O' = E. Similarly, because A ^ O then AA, = OA' ^ U. By 3.2.2.9, A' = Ay e K,' as 0,A e /C, /C is isotropic, and 0|y 1 a. Similarly, E' e K,' . Thus by 3.2.1.6 we have gEE, II goE' = IC' || g.,,,, and gEE, II g^^, by 3.2.1.7. By 3.2.1.8, h = gAA, and h n goE = gAA, r^ JC = {A}. Hence 5^(£) = A. 2 5^(£,) = ^,. Again, g8^(^E)A, = gAA, II gEE, and g^^, n / = {A,}. 3 Let P e X and P ^ O. U P ^ t then by 3.2.1.8, there is a unique fine g such that Et e g, g II ^4,p, and gngop = {Q}, for some Q e X. Then it follows that 5^(0 = P.UPet then 5a(Q) = P, where {^} = ^n /, £ e g, and g \\ gAP. ■ 62 Lemma 3.6.4. IfC ¥■ D, then gco II ^6^(C)8^(Z3)' hence, for all O ^ A e K, 6^ is a dilation ofX. Proof: Let g\ = t, g2 = goc- gi = gOD, P\ = Et, Q\ = A2 & g\. Pi = C, Q2 = 5^(0, Qi G ^2, ^3 = A and ^3 = 5..)(£>)- Then ^c£, II .?^,8^(o and gz)£, || gA,h^(Dy Thus, by Axiom D, gcD II gs^(08^{Z))- ■ Consider now the plane Xa- Recall that by 3.5. L3, iov O ^ A e IC, there is a unique 64 e VaiO) such that 5^(£) - A, where 6^ is of the form GgC^f, for g,h e Oa- From Axiom 9 and the proof of 3.5. LI, for any r e Oa, agCT/,CTr = Gw for a unique H- 6 Oa. That is, for every fixed r e Oa, every O ^ A e IC can be uniquely writen in the form 5^ = 0^^^ where Gw(E'') = A. (The uniqueness follows from the fact that (Ay^ = iA')''^ implies w^ ^ wj ii A' ^^ A [30].) Therefore, for every O ^ A e IC, there is a unique a e Oa, such that E'" = A. That is, there is a unique a e Oa, such that a^CT, = 5.,). Also, every P e Xa, P\oi, can be uniquely writen as P = P1OT2, where P\ e IC and P2 e IC'. Let P2 = P2' ^ fC. Then we may uniquely write P = P\0P2, where P\,P2 e /C. Define the map 6^ : .Xa ^ -^a, by 5^(P) = P'CO(P'f)', hr O ^ A e IC, P\a. Theorem 3.6.5. The map 8a : Xa ^ Xa is a dilation on Xa with fixed point O and dilation factor A, for all O ^ A e /C. Proof: If P'^OPf = QfOQ'f, then by unicity, Pf = Qf. So Pi = Q\, P'f = Q'f, and P2 = Qi- Hence, P ^ PiOP'2 = QiOQ'j = Q, and 5^ is injective. Let g|a with Q = Q\OQ'2, QuQ2 e K. Let P, = (?f e /C and P2 = 0' e IC. Then P = FiOP^Ia and 6^(P) = PfOPf = iQfy^OiQfY"' = Q,OQ\ - ^. Therefore, 6^ is onto. We claim that li P ^ O, then gpg || g%^(p)%^(Qy First we show that ^AiP) e go/'. If P 6 /C, then P = /'OO' and S^C?) = P'^OO"" - P'«00 = /"^ e AC. If P e IC', then putting R ^ P' e IC, we have P = 007?' and dA{P) = O'^OR"" = = OOR"" = /?'"' = P"' e IC'. 63 So suppose that gop - g is nonisotropic and write P = PiOPj, where P\ = P'f- UO e g, then XOY e g<^X= YS, where X e JC and Y e K.' . Thus, 5^(P) = PfOP'f and P'f = Pf = Pf^, which impUes that 5a(P) e g. Claim. dA(POQ) = 5^(^)05^(0, for ah P,Q e Xa- Let P = P^OP'2 and Q = Q\OQ\. Then POQ = P\OP\OQxOQ^2 = {.P\OQ\)0{P20Q2y, with P\OQx e /C and {P2OQ2)' e fC'. Therefore, ^a(POQ) = (PxOQxY^OiPiOQj)''' = PfOQfOP'fOQ'f = (PfOP'f')O(QfOQf) = dA(P)05A(Q). The claim follows. Proof of (iii): Assume that P,0, and Q are collinear. Then there is a line g, such that P,0,Q e g. Thus, 6^(P),6^(0 g g and gpQ = g \\ g = g^^^p)i,^^Qy Conversely, suppose that P, Q, and O are noncollinear. Let g\ = gop, gi = gOQ, and g2, = go.POQ- Then g\,g2, and ^3 are distinct lines through O. Indeed, if ^3 = gi, say, then POQ e gop and we would have OP(POQ) = Q e gop, which contradicts our assumption of noncoUinearity. Now, PiPOQ) = OQ and 6^(P)(5,4(/'O0) = 6^(6^(/')O6,,(0) = 06^(0 gPJ>OQ II gOQ = go6^(Q) II gh^iP)8^iPOQy ^nd gp^poQ || g8^(/>)5^(/.O0- Similarly, ^(^'00 = ^(eOP) = OP; and 6^(0(6^ (/'O0) = 6^ (6,4 (006^ (P)) = 05^(P). This implies that gg^pog \\ gop and gop = goh^p) II ^5^(05^(PO0- Hence, gQ,POQ II g5^(06^(/>o0- By Axiom D, it follows that gpg \\ g^^^p)^^(^Qy Therefore, 5a '■ Xa >—^ Xa is a dilation on Xa- B Now we show that 5^ = 6^ on Xa- Because 6^ and "6^ are dilations on aCa, a dilation is uniquely determined by the images of two points,and 6^(0) = O = 5^(0), then it suffices to show that 6,4 (£) = 6^(£). By definition of 6^, 5a(E) = A. By Lemma 3.6.3, 6^(£) = A. 64 To extend the above idea to any r| e A^ such that t c Xr\, let a 9^ r) e M such that / c Xr\. Let IC\ and AC2 be the isotropic Hnes in Xr] through O. Because / is nonisotropic, then by Axiom 14 there exist y,6 such that r) = y5 and / = [r|,y,5]. Thus (Tr : X,^ i-^- Xri is well-defined. Every B e t a Xr\ may be uniquely written as B = B\0B2, where 5, e /C,. Because O e t and t is nonisotropic, then B2 = B\. Thus, B = B]OB2 where B] & IC\. So, in particular, there are unique E\,A\ e IC\ such that Et = E\OE\ and At = A\OA\, for At e /.As before, there is a unique a\ e Ot) such that E"^ = A\. Hence, every X e X^ can be uniquely written as X - X1OX2, where X\,X2 e )C\ d Xr]. This defines a map 6^,^ : X^ 1— >• X^, given by dAr^iX) = X'C'OX'p' , for X e Xr^. Proposition 3.6.6. Let ^A^ : Xr[ ^ X^ be the map defined above. Then 1. 6^,1 is a dilation on Xr\. 2. 6^,1 = 5^ on X,y 3. ifP,0, and Q are collinear points in Xr\, not necessarily distinct, then bAy,{POQ) = 8Ar,(P)Od4r,(Q). 4. Moreover, Ia{POQ) = h(P)05AiQ), for every P,Q\y\ such that 0,P, and Q are collinear. U To obtain a scalar multiplication on V, for all O =^ A e IC and all OX e V, define A-OX= 6.4 (0)5,4 (A) = OdA(X) and 6> • OX = 1 ». We now verify the vector space properties. Lemma 3.6.7. If A, A' e /C and OP e V, then (A + A') - OP = A - OP + A' - OP. Proof: Suppose P e / and P ^ O, and recall /,/C c Xa. Let P,E\a,^; A\y with y || p; A'\y\ y' II p; and 0|5 with 6 || p. Then y || y' || 6; g, = [a,y] || gEP with g, c= Xa, 82 = [a,y'] II gEP with ^2 e .^a; and ^3 = [a, 6] || gEp with ^3 c .^a- Thus, bA{P) e gint and d^iP) e g2 n t. Thus, dA(P)\a,y and 5^iP)\a,y'. It follows that 65 dAiP)Od^iP)\a,y8y'; 5^(P)05^-(/') e t- y6y' = c || p; and ^O^' la^ySy' = s- Therefore, AOA' e [a,s] || ggp and 5^(P)06^'(P) e m [a,8]. Hence, B^oA'i^ = h(P)05^>iP). Suppose P ^ t. Because gopJ <^ ^ri, then replacing E with Ei, A with At, A' with /4„ and a with t] in the first part of the proof and the result follows. ■ Lemma 3.6.8. If A e IC and OP, OQ e V then A - (OP + OQ) = A • OP + A - OQ. Proof: We need to show that dA(POQ) = 5^(P)O6.4(0. Suppose P,0, and Q are coUinear. Let g = gop = goQ = goj'OQ and r\ & M such that g,t c ?i^. Then the result follows from Proposition 3.6.6(iv). Conversely, suppose that P, O, and Q are not collinear. Because P, O, and Q are not collinear then P ^ Q, and from Lemma 3.6.4 it follows that gpg \\ gg /mg ,Qy Applying Axiom R we obtain go,POQ = go5A{P)h(Qy '^^at is, 6^(/')O5^(0 e go,poQ by construction. Again by Lemma 3.6.4, 8h(P)UPOQ) II 80J>0Q and a.4(P)6.4(P)O5.,(0 = 05^(0. This implies that g5^(/>),5,(/>)o6,(0 II ^08,(0 = gOQ II gp,POQ, as P(POQ) = OQ. Thus, SdA(P),h(P)05A{Q) ^ .?5^(P)8^(/'O0 because both lines contain 5,j(P) and are parallel to gP.POQ- Since S5AP).h(P)08AQ)^SP.POQ = {5..,(/')C>5^(0} and g5A(P)h{POQ) ^ go.POQ = {5^(/'O0}, then 5^(P)O5.,(0 = 5^(PO0. ■ Lemma 3.6.9. IfA,A' e K and OP e V then A • (/!' • OP) = {A -A') • OP. Proof: We need to show that 5^(5^'(P)) = 5^.^'(P); that is, 5.^ o 5^- = 5^.^-. Now 5^ o 5^' and 6^.^/ are dilations on X and "5^.^'(0 = O by construction, so it suffices to show that (6,4 o 5.,0(£) = ^A.A'iE) and (5.^ o 5^,){0) = 5^.^'(0). Now £ e /C c JCa, so on .^a, 5.4 ° 5.,' = 5.4 o 5^, and 5^,y = S^.^-. But on /C, by Lemma 3.5.11, 5.4 ° 5^' = 5^.^'. Therefore, (5.4 o 3^>)(E) = A ■ (A' - E) = (A - A') • E = d^.^^^. ■ Lemma 3.6.10. For E e /C. the multiplicative unit, and OP e V. E - OP == OP. 66 Proof: We need to show that dg = l.x- Now Ix is clearly a dilation on X and 1 ^(O) = o = 6£(0). Thus, lx{E) = E = E'E = dsiE) = 5e(E). ■ Theorem 3.6.11. The space (V,)Q constructed above Is a vector space. ■ The triple (X,V,/C), is an affine space. [23, p. 6] A set X along with a vector space V over a field /C is an afBne space if for every v e V and for every X e 3£, there is defined a point vX e X such that the following conditions hold. 1. If v,!^ e V and X e X, then (v + w)X = v(wX). 2. If denotes the zero vector, OX = X for all X e X. 3. For every ordered pair (X, Y) of points of X, there is one and only one vector v e V such that vX = Y. The dimension n of the vector space V is also called the dimension of the affine space X. Theorem 3.6.12. (X,V,IC) is an affine space. Proof: If OV,OW e V and X e X, we have (i) {pV)X=OVX e X. (ii) (PV + OW)X = {OVOW)X = OV(OWX). (iii) 00X= 1<&X = X. (iv) for Y e X, OYX ^Z e X and iOZ)X = Y. Now if OPX = Y, then OPX = OZX, P = Z, and OP = OZ. ■ 3.7 Subspaces and Dimensions In this section we show that our lines and planes have the proper dimensions. We are then able to conclude that (V,/C) and (ie,V,/C) are four-dimensional spaces. Proposition 3.7.1 Let g be any line through O and put giO) = {OA : A e g}. Then giO) Js a one dimensional subspace ofV. Proof: First note that !& = OO e g{0), so the zero vector is in g(0). Let A,B e g. Then C = AOB e g hy 3.1.4.7 and OA + OB = OAOB ^ OC e giO). From Section 3.6, 67 5^(5) e g, for all A in K. and all B in g. So A • OB e g(0) for all ^ e /C and OB e g{0). Hence, g(0) is a subspace of V. It must now be shown that the dimension of g{0) is one. li g = t, then giO) = (OEt) because for every At e t, Ai - 6^(£;) by Lemma 3.6.3. So suppose that g =^ t and fix B ^ g. Let h - gBE,- Then for slX O ^ D & g, there ia a unique rfsuch that D e d, d \\ h, and drM ^ 0. Put {Ft] = dnt. Then for F, = FOF' with F e IC it follows that 6f-(5) = D and OD = F - OB. Hence, ^O) = <05). ■ Corollary 3.7.2. Following the terminology of Snapper and Troyer (23, p.U], g = S{0,g{0)) ^{A = 0{0A) : OA e g(0)} is an afSne subspace of dimension one. Proposition 3.7.3. Let a e V with 0|a and put tai.0) = {OA : A\a}. Then taiO) is a two dimensional subspace ofV. Proof: Clearly, 1© = OO e taiO), so the zero vector is in ta(0). Let C,D|a and A.B e IC. Then by 3.L4.7 we have COD = F|a and by Lemma 3.6.3, 5/1 (O e goc cz Xa and 6b(D) e goo c= Xa so that dA(QOdBiD)\a. It follows that OC + OD = OCOD = OF e ta(0), and A-OC + B'OD = O'dAiQ + OdBiD) = 0(6^(0055(D) e .ta(0). Hence, ^aiO) is a subspace of V. Thus, it remains to show that ta(0) is two dimensional. There are two cases: a e >1 and a e g. Suppose first that a e A^. We construct a basis for ta(0) using isotropic lines. To this end, let /Ci and /C2 be the isotropic lines in .^a through O. For any Pla we may uniquely write in Xa, P = PxOPj, with Pi e /Ci and P2 e Ki- 68 From Proposition 3.7.1 above, ^i(O) = {OB) for any O ^ 5 e /Ci and K.2(,0) = (OQ for any O t^ C e ICj. From this it follows that P] = 6^(5) and Pj = §^'(0 for some A, A' e /C. Thus, OP = A -08 + A' 'OC and {OC,OB} span ta{0). Now \i A - OB + A' - OC = 1©, then C>5^(5)06^'(Q = I©. This implies that 6^(fl)05^'(C) = O and Ol^^{C) = 5^(5)0. Because 0,5^'(C) e /C2 and 0,5^(5) e /C,, then either /Ci || Kj or (95^'(Q = 1© = 6^(fi)0. But /Ci ]f /C2, so 5^'(0 = 5.4(5) = O. Because a e A1, then from 3.3 and 3.6 there exists t' ,a e Oa such that 5^'(0 = C'"' = O. This implies that C = O"''' = O ov A' = O. By assumption C * O and D ^ O, thus it must be the case that A = A' = O. Hence, {OC,OB} is a linearly independent set in %a(,0) and -ta(0) is two dimensional. Suppose now that a e Q. We construct an "orthogonal" basis. By 3.1.7.2, there exist p,Y e ^ such that O = apy and a 1 P 1 y 1 a. Let jc = [a,p,aP] and y = [a,y,ay]. Note that if ;c = _v then by Axiom 14, aP = y and O = aPy = 1®, so x^y. Let P| a and let p',y'|/'with p' 1 p and y' 1 y. Put P, = pp' and P2 = yy'. NowP|a,p'; a 1 p; p' 1 p, so that a 1 p' by 3.1.6.18. Because P| a, y'; a 1 y; y' 1 y, then a 1 y'. Thus, P? = (PP')« = P«P'« = pp' = P, and Pf = (yy')« = yy' = Pj, so P\,P2\a.. This implies that P\ |a,P and P2|cx,y so P] ex and P2 e y. If Q = PxOP2 then Q ^ P\OP2 = P'ppayyy' = p'ay'. Because p' 1 a, let § = ap' e V. Then Q = 5y' and 5 1 y'. Thus P| p',a implies that P| p'a = 5. Thus we have P,Q\d,y' with 6 1 y', and P = ^ by Axiom 3. That is, P = P1OP2 with P] e x and P2 ey. Let O ^ X e X and O :^ Y e y. Then from Proposition 3.7.1 above we have X = (OX) and y = (OY) and there exist A,B e IC such that OP = A • 0X+ B • 07, where 69 Pi = 6a(X) and P2 = 8b{Y). Hence, {OX,OY} spans .ta(0). li A • OX+ B - OY ^ 1©, then we obtain O5.4(J0 = ^b(Y)0. Because x J[ >», it follows that ^^{X) = O = 85(7)- Let ^ti be the unique plane containing t and ;c and K,\ and K.2 the isotropic lines in Xri through O . Then we may write §4(^0 = X'^OX'^', where X^ e /Ci, and X^ e IC2. As above it follows that A = O and similarly, B = O. Therefore, ^aiP) is two dimensional. ■ Corollary 3.7.4. Xa = S(0,!^a(0)) Is an affme subspace of dimension two for all a e P, 0|a. Theorem 3.7.5. (V,/C) is a four-dimensional vector space and lience, (X,V,}Q is a four- dimensional affine space. Proof: Let OP e V and let O = ap with a e X and p e ^. Let P|a',p' with a' 1 a and p' 1 p. Put P' = a'a and P" = p'p. Then Q = P'OP" = a'aappp' = a'p'. Thus, P,e|a',p' with a' 1 p'. Therefore P = Q = P'OP" with P'\aand P" | p. Since taiO) and .tp(0) are two-dimensional then there exist bases {OZ,OT} c .ta(0) and {OX,OY} c= tp(0) such that OP' =A'OZ + A' • OT and OP" = B • 0^-+ B' • 6>y for some ^,^',5,5' e IC. Thus, P = ^ -02 + y4' -OT+B-OX+B' • 07 and {OX,OY,OZ,OT} span V. If ^ • 0Z + ^' • 07+5 • (9X+ 5' • 07 = I©, then in particular, OP = OP' + OP" = 1©. This implies that OP" = P'O. So either Sop" II ^/"o or P" = P' = O. But ^^p" if gop' and therefore, A '0Z + A' -07= OP' = 1& = OP" = B'OX+B' 'OY. As was shown in Proposition 3.7.2, we obtain ^ = ^' = 5 = 5' = O and the result follows. ■ 3.8 Orthogonality In this section we extend the definition of orthogonality to include lines and then use this definition to define orthogonal vectors. 70 Definition 3.8.1. Let g and h be two lines. We say that g is perpendicular to or orthogonal to h, denoted by g J. /?, if there exist a,p e "P such that g c Xa, h c Xp, alp, and P = a^ ^ g,h. In this case, P is the point of intersection of g and h. Lemma 3.8.2, Ifg and h are isotropic then g is not orthogonal to h. Proof: This follows directly from our definition above and from our definition of alp. For if a 1 P then one of a and p must be in Q and by Axiom 12, all lines in a plane J£p for P e Q are nonisotropic. ■ In Minkowski space, if g and h are two isotropic lines then g L h ■(r^ g \\ h. Thus we extend the above definition in the following way. Definition 3.8.3. If g and h are isotropic lines, then g L h ■(r^ g \\ h . Definition 3.8.4. If g is a line and a e P then we say that g is orthogonal to or perpendicular to Xa, g 1 -^a, if there exists a P e T' such that g (= JEp, P 1 a, and P = ap e g. In this case, P = ap is the point of intersection of g and Xa- Definition 3.8.5. For every 1® ^ OAMB e V, we say that OA is orthogonal to OB, OA 1 OB, if and only if go^ 1 gos; that is, there exist a,p e P such that goA c Xa, gOB c Xp, and O = ap. For the zero vector 1(5= OO, we define !<& L OA, for all OA e V. Lemma 3.8.6. Fronj 3.1.2.1 and 3.1.2.2 it follows that for?, e 6: {ij g 1 /? <^ g^ 1 h^. (ii) gl Xa "^g^ 1 X^t. (Hi) OA lOB^ (OA)^ 1 (OB)^. ■ Lemma 3.8.7. Ifg is a nonisotropic line then g is not orthogonal to g. Proof: If g 1 g then there exist a,\^ e V such that g c Xa,Xp; alp, and P = ap e g. But for every point Q e g, Q\a,^ with alp which implies that P = Q by Axiom 3; that is, g is a line which contains only one point, which contradicts the definition of a line. ■ 71 Additional axioms and their immediate consequences. To complete our preparations for defining our polarity and thus obtaining the Minkowski metric, we recall our final three axioms. Axiom U. (U^ subspace axiom) Let 0,A,B, and C be any four, not necessarily distinct, points witli A,0\a; 0,5|P; 0,C|y,6 and a 1 y and ^ 1 5. Then there exists ?.,s e V sucli thatXl e; OAOB\'k\ and 0,C\z. Axiom SI. Ifg cz Xa, a e ^, /z c .tp, (3 e Q, and there exists y,5 e V such that y 1 5; yd e gnh; g cz Xy-, and h c .^5 then there exists e e g such that g,h c Xg. (If g and h are two orthogonal spacelilie lines then there is a spacelike plane containing them.) Axiom S2. Let g and h be two distinct lines such that P e gnh but there does not exist ^ e V such that g,h c .^p. Then either there exists a,y 6 7^ such that a 1 y, g (= Xa, andh c Xy or for all A e g, there exists B e h such that P and APB are unjoinable. Lemma 3.8.8. Ifg,h cz Xa are nonisotropic for a e V, then g 1 h in the sense of Section 3.3 if and only ifg 1 h in the sense of Definition 3.8.1. Proof: Let g = [a,p,6], h = [a,y,?L] c Xa with a = (35 = yX. Recall that g Ihin the sense of 3.3 if , without loss of generality, P 1 y and A ^ ^y e g r\ h. So, in particular, A = ^y with A e g n h and g cz Xp and h c Xy. So g 1 A by 3.8.1. Now suppose that there exist r|,s e 7^ such that B = r\e e gnh and g (^ X and h c Xe, but it is not the case that ply, nor that p 1 A, nor that 5 1 y, nor that 6 1 A. Then by 3.3.8 there exists a unique / e £„ such that B e I and I = [a,u,^] for some u,^ e P with a = u|i, u 1 p, and |a 1 5. If / 9^ /z then /a(5) and ha(B) span .ta(5). Thus ii C e g there exists L e I and H e h such that BLBH = BC. By Axiom U, BL 1 BC and BH 1 BC imply that BC = BLBH 1 BC. That is, gig, which cannot happen for nonisotropic g. Hence, I ^ h and the result follows. ■ 72 Lemma 3.8.9. If a e V then for each P \ a and for each nonisotropic line g c Xa there is a unique nonisotropic line h a Xa such that P e h and h L g. Proof: This follows directly from 3.3.8 and Lemma 3.8.8. ■ Lemma 3.8.10. Ifg,h c Xa for a e V and g,h are nonisotropic, then g -L h if and only if (J gO I, = CT/,ag ^ l.i-„. Proof: This follows from 3.3.16 and Lemma 3.8.8. ■ Lemma 3.8.11. Suppose that O e c,d; c,d (^ Xa, ex e V with c and d nonisotropic and c -L d. Let C e c and D e d, then go,DOC is not orthogonal to c ifC,D ^ O. Proof: First note that go.DOC "^ c ov d because then we would have DOC = D\ e d, say, so that C = ODD\ e d and O e d implies that C = O or c = d. Now if go.DOC -L c, then in Xa we have Gq = cJcG^ = (yg(,j-,^x.(Jc, which implies that d = gO,DOC- ■ Lemma 3.8.12. Let g,x c Xa with g isotropic, a e V, and g n x = {O}. Then g is not orthogonal to x ifg ^ x. Proof: If X is isotropic and x 1 g then because x ^ g, there exists y,d e V such that g a Xy, X c Xs, and y J_ 6. But then one of y and 5 must lie in Q. But no element in Q can contain an isotropic line so x is not orthogonal to g. Suppose that x is nonisotropic and let O e g D x. Because g is isotropic then a e M and by 3.3.8, there is a unique nonisotropic line h (^ Xa such that O e h and h Ix. Suppose that g ±x and let O ^ K e g, O ^ H e h, and O * X e x. Then OH 1 OX and OKI OX so that by Axiom U, 0{HOK) = OHOK 1 OX. li go.HOK is nonisotropic then go.HOK = h. And because go,HOK c= Xa, then HOK = //, eh. Then we have K = OHH\ e h and g = h, a contradiction. Suppose that goMOK is isotropic. If goMOK = g then we may write HOK = Kx for some Kx e g. It follows that H = KiKO e g and g = h. li go.HOK * g then goMOK is the other isotropic line through O in .^a- Because g and go,HOK span Xa and 73 g,go.HOK -L X then by Axiom U, x is orthogonal to every Hne in Xa through O. So in particular, x -L h^ which implies that x = h. ■ Corollary 3.8.13. Ifg Js isotropic and x L g tlien either x - g or x is nonisotropic and X and g are noncoplanar; that is, there does not exist 8 e V such that x,g c X5. Lemma 3.8.14. Ifg is isotropic, x is nonisotropic, x L g with x ^ g, and {P} = x H g, then for all P * A eg and for all P ^ B e x, g and gAPB are not coplanar. Proof: From Lemma 3.8.12, x and g are noncoplanar. Suppose that A,0,AOB |y for some y e V. Then B = OAAOB\y, which implies that x = goB <= Xy and g = goA <= Xy, a contradiction to Lemma 3.8.11. ■ Lemma 3.8.15. IfOC e V is isotropic and OB e V is nonisotropic with OC 1 OB then OCOB 1 OC. Proof: By Lemma 3.8.13, goc and goB are not coplanar and by Lemma 3.8.14, goc and go,coB are not coplanar.By Axiom S2, either goc -L gocoB or there exists D e gocoB such that O and COD are unjoinable. So if goc is not orthogonal to gocoB then go.coD is isotropic and by Axiom T, there exists a unique d e V such that gocgo.coD c= -^6- This implies that 0,C,COD\d, D = OCCOD\8, gOD = go.coD c= -^8, and therefore, goc <= -Xg, a contradiction. ■ Lemma 3.8.16. The zero vector, 1© = 00, is the only vector orthogonal to every vector in V. Proof: This follows immediately from Lemma 3.8.11 above. ■ Theorem 3.8.17. IfU is a subspace ofV, then U^ = {OA e V : OA 1 OB, VOB e U} is a subspace ofV. Proof: because the zero vector 1© is orthogonal to every vector in V by definition, then 1© &U^. Let OA e U^ and /? e /C. because R»OA e< OA >, the subspace generated by OA, and goA 1 gOB for every OB e U by the definitions oW^ and orthogonal vectors, then R»OA e U^. 74 Let OA,OB e U^ and OC e U. If OC is nonisotropic then OC ^ OA,OB and there exist a,P,Y,6 e V such that A,0\a; 0,5|p; 0,C|y,6; a 1 y; and p 1 5. By Axiom U, there exists 'k,e e V such that O = A,s, AOB\X, and 0,C |e. Thus, OA + OB ^ 0{AOB) 1 OC. Now suppose that OC is isotropic. The following possibilities exist. (i) If OC * OA,OB, then as in (a) above, (OA + OB) 1 OC. (ii) If OC = OA = OB, then because OC is isotropic, AOB = COC e goc and {OA + OB) 1 OC. (iii) If OC = OA It OB, then OB _L OC implies that OB is nonisotropic and (OC+OB) = OCOB 1 OC by Lemma 3.8.14. Hence, if OA,OB e U^ then OA + OB e U^ and U^ is a subspace of V. ■ Theorem 3.8.18. IfO = ap then ta{0)^ = .tp(0) and t^{0)^ = taiO). Proof: From the definition of orthogonal vectors we clearly have ta(0) a .tp(O)-^ and tp(6>)^ c ta(0). Now suppose that OA 1 .ta(0); that is, goA 1 a. Then, there exists y e V such that OA e ty(0) and y 1 a. But then we obtain (9 = aP = ay so p = y. Hence, taiO)^ = -tp(0) and ta(0) = -tp(0)^. ■ An immediate consequence of Theorem 3.8.18 is the following. For each a e V with 0\a, there exists a unique ^ e V such thatO|p, taiO)^ = ^fi{0), and Theorem 3.8.19. Ifd{0) js nonisotropic then there is a unique hyperplane AiO) such that d{0)^ = A{0) and (fl(0)^)^ = d(0). Proof: Let a e V such that a a Xa, where a is the nonisotropic line associated with d{P) = {OA : A & a]. Then for O = ap we have a ± p and o 1 g for every g c Xp with O e g. Thus, .tp(0) c ^(O)-^. Now a is nonisotropic, so by 3.3.8, there exists a unique h c .^a such that O e /? and a 1 /z. By Axiom U, </?(0),tp(0)> c d{0)^. 75 Suppose l(s ^ OB e d(0) n (h(0),tpiO)).Then B e a a Xa and there exists OC e h(0) and there .exists OD e .tp(0) such that OB = OCOD; that is, there exists C e h and D \ p such that B = COD. because h c Xa, Qa and C = aa|. Because Z)|p, D = pp, for some p, e 7^. Now a 1 p and p 1 p,, so a || Pi. But B = COD and COD = ajaaPpPi = aipi so that 5|pi,a with a || pi. This imphes that a = Pi so D = O, B = C, and a ^ h, a contradiction. Hence, d(0) n (h{0),tpiO)) = {!©} and V = d(0) © </?(0),.tp(0)>. U O e d and d 1 a then diO) c </7"(0),.tp(0)>. For otherwise we would have V = d(0) ® d(0) @ </7"(0),tp(0)>, which is not possible. Hence, d{0)^ - </z(0),.tp(0)>. On the other hand, from .tp(0)^ = -ta(0), a 1 h. and a ± p, then, d(0) c (h(0),ta(.0))^. If O e ^ 1 p, then by definition g ^ Xa- H O e g 1 h, then g = a. Hence, d(0) = {h(0),XpiO)}^. Claim. d{0)^ is independent of the plane containing a. Suppose a cz Xy ^ Xa- Then for O = y5, we have y ^e a, 5 ^ p, and a 1 X?,. Again there exists a unique / c Jy such that O e / and / 1 a, so that d(0)^ = </((9), -t5(0)> as above. Now any point P e X may be written as P = PxOPi with P] |a and Pj \ p. Since h ^ a, then we may write Px = //O^ with H e h and ^ e o. Then <MO),tp(0)> ®d{0) = V= </(0),.ty(0)> ©«(0), and it follows that </j"(0),.tp(0)) = </(C>),.ty(0)> = A{0). ■ Theorem 3.8.20. Ifd{0) Is isotropic tlien there is an unique hyperplane A{0) such thatdiP)^ = A{0),d{0) d A(0), andidiO)^)^ = d(0). Proof: Let d(0) c taiO) and a the isotropic line in Xa corresponding to d(0). because a is isotropic then a e A1. Let O = ap, p e ^, and put A(0) = (a(0),.tp(0)>. Since a c Xa, alp, and O = a^ e a, then a 1 Xp. Because 76 a is isotropic then a 1 a. By Axiom U and Lemma 3.8.14 it follows that <d(0),tp(0)> = A{0) c d{0)^. lig{0) c <d(0),tp(0)) is isotropic and g ^ a, then there exists a unique y & M such that g,a a Xy. Because g(0) c A{0) a d(0)^, then g ± o, which contradicts Lemma 3.8. U. Thus, AiO) contains no other isotropic line. Let h 1 a with O e h. For each H e h we may write H = H' OB, with //'|a and 5|p. If O ^ ^ e o, it follows that OB 1 O^; O// 1 OA, so that 05^ 1 OA and O//' = 0//C>5^ 1 OA. This implies that gg^^ = a so H' e a and /2(0) e <d(0), j&p(0)>. Thus, fl(0)^ = <a(0),-tp(0)). If //'(O) c <o(0),tp(0)>^, then hiO) c .tp(0)^ = .ta(0) and h(0) 1 d(0). Thus, h(0) = diO). Hence, <d(0),.tp(0)>^ = d(0) and (d(0)^)^ = d(0). The subspace ^(O)-^ is independent of the plane containing a. Suppose a cz Xy, y e M.y^a.FutO = y6. Let h(0) c {d(0),t8(0)). Then /? 1 a and ^((9) c (d{0),tp(0)). If ^(O) c <d(0),.tp(0)> then A: 1 a and k(0) c <fl(0), ^5(0)). Therefore, <d(0),.t8(0)> = <d((9),tp(0)>. ■ Remark. If g is a line then g is either nonisotropic or isotropic. From Theorem 3.8.19 and Theorem 3.8.20, if W < V is a one-dimensional subspace then U^ is a uniquely determined hyperplane. In an affme space a plane is uniquely determined by two distinct intersecting lines. Let g and h be two distinct intersecting lines and let {g,h) denote the unique plane determined by g and h. 77 Definition 3.8.21. U a e V such that g,h a Xa then we say that (g,h) = Xa is a noDsingular plane. If there does not exist a e V such that g,h c: Xa then we say that {g,h) is singular or (g,h) is a singular plane. It is clear that every plane in {X, V, /C) is either singular or nonsingular. Note that by Theorem 3.8.18, ifiY<Visa nonsingular two-dimensional subspace of V, then U^ is a uniquely determined nonsingular two-dimensional subspace of V. Now consider the following cases for two distinct intersecting lines g and h. Suppose that g is isotropic and h is isotropic. Then by Axiom T, there exists a unique a e M such that g,h cz Xa. If one ol g ov h is timelike, then by Axiom T, there exists a unique a e M such that g,h a Xa- Thus, if {g,h) is singular then either g is isotropic and h is spacelike, or g and h are both spacelike. Proposition 3.8.22. Let {g,h) be a singular plane with g isotropic and h spacelike. Then g L h. Proof: Let {P} =^ gnh. By Axiom S2, either g 1 h or for each A in g, there exists B in h such that P and APB are unjoinable. Suppose that A e g, B e h, and P and APB are unjoinable. It must be the case that A is joinable with APB; that is, gAjPB is nonisotropic. Let gPA = [6,e], so A,P\5,s. If A is unjoinable with APB then A, P, and APB are pairwise unjoinable points, so by Axiom 10, APB\b,z and APB e g. Thus, gPjPB "= gPA = g and B = PAAPB eg, so g = h. Thus, gpjPB ^ g and gpjPB is isotropic, so by Axiom T, there exists a unique y e M such that g,gpjPB c Xy. This means that P,A,APB\y so B = PAAPB\y, and h c Xy, which contradicts our initial assumption. Therefore, g ± h. ■ Theorem 3.8.23. Let {g{0),h(O)) be a two dimensional singular subspace ofV where g{0) is isotropic and h(0) is spacelike. Then {g(0),h(0))^ is a two dimensional singular subspace of V which contains giO). Moreover, i{g(0),h(0)}^)^ = (g(0),h(0)). Proof: First we observe that if W is a subspace of V generated by subspaces U and U' 78 then W^ = {U,U')^ = U^nU'^, v 6 {U,U')^ <^ v e U^, and V eU'^ -^v eU'^ n U^. Consider (g(0),h(0))^ = g(0)^ n h(Oy. Because g Ihhy Proposition 3.8.22 then there exists y,d e V such that (9 = y5 with g c Xy and h c .^6. From Theorem 3.8.20, giO)^ = <i(0),ts(0)) and by Theorem 3.8.19, h(Oy = </(0), Xy(0)) where / c .^5 is the unique line through O in .X§ orthogonal to /?. Hence, {g{0),hiO))^=g(0)^nh(0)^ = <^(0),-t6(0)) n</(0),ty(0)) = <^(0),/((9)>. Moreover, {g(0),kO))^ = g(0)^ nl(0)^ = (giOltsiO)) n{hiO),XyiO)) = (g(0),hiO)). Now if g,/ c .^e for some e e V then by Theorem 3.8.18, for O = sp, {giO)J(O)) = .ts(0) and <^0),/f(0)> = (g(0)JiO))^ = ^^0)^ = .tp(0). This says that g,h a Xp; that is, {g,h) is nonsingular. Thus, <g((9),/(0)> is nonsingular. If {g,h) is a singular plane, O e gnh, and g and /? are spacelike, then by Axiom SI, g is not orthogonal to h. So by Axiom 2, for all yi e g, there is a B e h such that O and AOB are unjoinable. ■ Theorem 3.8.24. For the above setup: 1. go JOB e {g,h). 2. go JOB -L g,/?. 3. ifCeg andD e h such that go,coD is isotropic then go,coD = go job- 4. (g{o),hio)y = {gojoB(o),giO)y. Proof: For 1., if there exists y eV such that gojOB,g c Xy, then a^,v405||y implies that B = 0^0 Iy and h c .Xy. This yields (g,h) - Xy is nonsingular. For 2. 79 go JOB -^ g,hhy Axiom S2 because by 1. above go,coD <= {g,h) and {g,h) is singular. lie & g and D e h such that go,cOD is isotropic then by l,.go,coD,gojOB c (g,h). If go,coD '^ go JOB then by Axiom T there exists a unique r\ e V such that g0,C0D, gOJOB <= -^y- Because two intersecting lines uniquely determine a plane, then (g,h) = ^Ey Hence, go,cOD = go JOB- The last conclusion follows from <^,/7> = (g, go job)- ■ If <g,/2> is a singular plane then from the proof of Proposition 3.8.22 and from Theorem 3.8.24, it follows that for each P e {g,h) there exists a unique isotropic line I (z {g,h) such that Pel; that is, {g{0),h{0)) contains one isotropic line. To complete the classification of orthogonal subspaces of V, it remains to consider hyperplanes. Now if {?i, V, /C) is any four dimensional affme space then a line and a plane which intersect in a point uniquely determine a hyperplane and any hyperplane in the space can be characterized as the subspace generated by a corresponding line and plane. Let A = </?, p> be the hyperplane generated by a line h and a plane p which intersect in a point O. Then From Theorem 3.8.19 and Theorem 3.8.20, we know that the dimension oi h(0)^ is three. By Theorems 3.8.18, 3.8.23, and 3.8.24 it follows that the dimension of (HO)^) = 2. Consider the following possibilities. If h(0)^ n $(0)^ = {!&}, then h(0)^ © p(Oy c V has dimension five whereas the dimension of V is four. If h(Oy n p(0)^ = p(0)^, then it follows that p(0)^ c hiO)^; h(0) 1 p(0)^ and KO) e (Hoy)^ = p(0). Which contradicts the assumption that h and p intersect only in O. 80 Therefore, h(0)^ n piO)^ - g{0) for some line g containing O. Because g must be either isotropic or nonisotropic, then the following is true. Theorem 3.8.25. IfA(0) js a hyperplane the there is a unique line g such that A{0)^ = giO) andiA(0)^y = A(0). ■ 3.9 The Polarity In this section a polarity is defined so that we can obtain the metric through a process given by Baer [3|. For the convenience of the reader, the pertinent definitions and theorems [3] are given below. Definition 3.9.1. An autoduaJityn of the vector space V over the field /C is a correspondence with the following properties: 1. Every subspace U of V is mapped onto a uniquely determined subspace n(U) of V. 2. To every subspace W of V there exists one and only one subspace W of V such that 7t(W) = U. 3. For subspaces U and W of V, .U < W if, and only if, Ti(W) < 7r(ZY). In other words, an autoduality is a one-to-one monotone decreasing mapping of the totality of the subspaces of V onto the totality of the subspaces of V. Definition 3.9.2. An autoduality n of the vector space (V,/0 of dimension not less than two is called a poJarity, if n^ = 1, the identity. Definition 3.9.3. A semibilinear form over (V,/C) is a pair consisting of an anti-automorphism a of the field K, and a function y(x, y) with the following properties: (i) y(x, y) is, for every x, y e V, a uniquely determined number in /C. (ii) Xa + b, c) =y(a, c) +/b, c) and y(a, b + c) =./(a, b) +/(a, c), for a, b, c e V. (iii) y(/x, y) = (/(x, y) andy(x, /y) =./(x, y)a(0 for x, y e V and t e K. If a = 1 then /is called a bilinear form. Definition 3.9.4. If/is a semibilinear form over (V,/C) and if Z^ is a subset of V, then 81 {x e V : fix, u) = for everyu e U} and {x e V : J{u., x) = for every u e ^} are subspaces of V. We say that the autoduahty tt of (V, /C) upon itself is represented by the semibilinear form fix, y) if n{U) = {x s V -.fix^U) = 0} = {x e V :y(x, u) = for every u e U}. Theorem 3.9.5. [3] Autodualities of vector spaces of dimension not less than 3 are represented by semibilinear forms. ■ Theorem 3.9.6. [3] If the semibilinear forms f and g over (V,/C) represent the same autoduahty ofV, and if dim(y) > 2, then there exists aO ^ d e K. such that ^x, y) = Xx, y)d for every x, y e V. ■ Definition 3.9.7. If n is an autoduahty of the vector space (V,/C), then a subspace W of V is caUed an N-subspace of V with respect to tt, if (v> <7i«v)) for every v e W. In this case, 7t is said to be a null system on the subspace W of V. Theorem 3.9.8. [3] Suppose that (f,a) represents the autoduahty n o/(V,/C). Then n is a null system on the subspace W <r^fiw,w) = for every w e W. M Definition 3.9.9. A fine <v) is called isotropic if (v) < 7i«v». (So an isotropic line is an N-line.) Theorem 3.9. 10. [3] If the semibilinear form (f,a) represents a polarity n, and if fiw,w) = 1 for some weV, then a^ = 1 and ^{Kx,y)) =J(y,x) for every x,y eV. In this case we say that/ is a -symmetrical or Just symmetrical ■ Theorem 3.9. 11. [3] Suppose that n is an autoduahty of the vector space (V,]C) and that dimiV) > 3. Then n is a polarity if, and only if, % is either a null system or else n may be represented by a symmetrical semibilinear form {f, a) with involutorial a. ■ Theorem 3.9. 12. [3] Suppose that the polarity n of the vector space (V,/C) possess 82 isotropic lines and the dini(V) > 3. TJien n may be represented by bilinear forms if, and only if, (a) planes containing more than two isotropic lines are 'H-planes and (b) K, is commutative. ■ Theorem 3.9.13. Suppose that n is a polarity of the vector space (V,IC) such that the conditions of Theorem 9.12 are met. Then from Theorem 3.9.11 it follows that ifn is not a null system then n may be represented by symmetrical bilinear forms. ■ Defining the polarity n and obtaining the metric g. Consider (V,/C) constructed in this work. liU is any subspace of V then by Theorem 3.8.17, U^ is also a subspace of V. From the end of section 3.8, if W is a subspace of V, ZY 9^ {I©}, and U ^ V, then U-^ is a uniquely determined subspace of V. From Lemma 3.8.11 and Definition 3.8.5, {l©}-^ = V and V-^ = {I©}- So we define the mapping n on the subspaces of V as follows: Definition 3.9.14. If ZY is a subspace of V then tz{U) = U^ . From the remarks above and from Section 3.8 it is clear that to every subspace UoiV there exists one and only one subspace W of V such that Tt(>V) = U. Theorem 3.9.15. Let U and W be subspaces ofV, then U < W if, and only if, n(W) < n{U). Proof: Suppose that U < W. U OV e W-^ then OV 1 OW for every OW e W, U cz W, so OV 1 OU for every OU e U and OV e U^. Thus, ZY < >V implies that k(>V) = W^ <U^ = n{U). Conversely, suppose that W^ < U^. By Lemma 3.8.11 and Definition 3.8.5 we have ({1©}^)^ = {1©} and (V^)^ = V. From the previous section it follws that, if ZY is a nontrivial proper subspace of V then (U^)^ = U. because W^ and U^ are subspaces, then from the first part of the proof we have W^ < U^ implies that U = QA^)^ < (W^)^ = W. ■ Corollary 3.9.16. From Defmition 3.9.14 and Theorem 3.9.15 above it follows that n : U < V ^U^ <V is an autoduality. Moreover, because QA^)^ =U for every 83 subspaceU ofV, then % is a polarity. Lemma 3.9.17. Let^ e (& and OA e V. Then niU^) = n{U)^ for every 'i, e © and for every subspace U <V. That is, 7t is invariant under cj^ for every ^ e 0. Proof: We have 0{OAf- = 00^ A' = B e X, so (OA)' ^ OB e V. Hence, from Lemma 3. 8. 6. the result follows. ■ Theorem 3.9.18. The polarity n is not a null system. Proof: Let a be any nonisotropic line with O e a. By Lemma 3.8.7, a is not orthogonal to itself so that d(0) $ d{0)^ and hence, d{0) $ n{d(0)). By Definition 3.9.7, 71 is not a null system. ■ Theorem 3.9.19. The polarity n defined above may be represented by bilinear forms. Proof:. Let g be any isotropic line through O. By Definition 3.8.5, g{0) < giO)^ so that g{0) < nigiO)) and g{0) is an isotropic line in the sense of Definition 3.9.9. Conversely, if /z is a line such that h{0) < n{h(0)) = h{0)^ then h 1 h and by Lemma 3.8.7, h must be isotropic in the sense of Section 3.2. Thus, (V,/C) possesses isotropic lines. By Proposition 3.7.3, dim(V) = 4 > 3. From Definition 3.8.21 any plane in (V, /C) is singular or nonsingular. Now a singular plane contains only one isotropic line by Theorem 3.8.24. If Z^ < V is a nonsingular plane then U = .ty(6>) for some y e V. li y e M then U has precisely two isotropic lines, as was shown in 3.2.2.8. If y e ^, then by Axiom 12 and Section 3.2, U does not have isotropic lines. Thus, no plane of V contains more than two isotropic lines. Since the field /C = R is commutative then by Theorem 3.8.12 the result follows. ■ Theorem 3.9.20. The polarity n may be represented by symmetrical bilinear forms; that is, there is a symmetrical bilinear form g such that for any subspace U ofV, U^ = n{U) = {OA eV: giOA^U) = 0} or g(OA, OB) = = giOB, OA)if, and only if, OA 1 OB for OA, OB e V. Proof: This follows directly from Theorems 3.9.18, 3.9.19, and 3.9.13. ■ 84 Thus, we have a metric g, a symmetric bihnear form, induced by our polarity, which agrees with our definition of orthogonal vectors, which in turn is induced by and is defined in terms of the commutation relations of the elements of our generating set Q. Lemma 3.9.21. Let g be a symmetric bilinear form representing n. Then g is nondegenerate. Proof: This follows from the fact that 7i(V) = V-'- = {l©}. ■ Theorem 3.9.22. Let g be a symmetric bilinear form representing n. Then there is a basis ofV such that the matrix ofg with respect to this basis has the form ^ C ^ C C V -c ^ C e IC: J that is, there is an orthogonal basis ofV such that the matrix ofg with respect to this basis has the above form. Proof: Let a e yW such that /C c .^a and let O = ap. Let 0|y,5 e Q such that a = 75 and y,5 1 p. Let s = yp e >! and r| = 5p e A^.Then zr\ = ypp5 = y6 = a so s 1 T). Put X = [a,y,5], y = [p,y,s], z = [p,Ti,5], and t = [a,s,Ti]. Claim. x,y,z,t are four mutually orthogonal nonisotropic lines through O. We have O = ap, O e x,t a Xa and O e y,z a Xp, so that xj 1 y,z. Also, O = aP = y5p = yr| = 58 so that y 1 t] and 6 1s. Then O e X cz Xy,0 e t a X^,0 e Xy, and O e z c Xr\ implies that x J. / and y ± z. The construction of the basis. Let E e fC, the multiplicative identity, considered as a point in .^a- Put T = EOE' e t and X = EOE'' e x. Note that because 85 O e x,t cz Xa and x 1 t then OxOi = gq = <7;a.r in Xa- Now t cz Xz and s e M, so there exist precisely two isotropic Unes, /Cig and ICje, through O in Xe. Thus, there is an unique Ee e /Cig such that T = EzOE'^. Because y c Xz then 7 s E^OEl e j; and CTya, = ao = CT/CTy in .^g. Similarly, because t cz X^, r\ e M, there are precisely two isotropic lines, K\^ and /C2T1, through O in .^ti and there is £,1 in K-\^ such that T = Er^OE'y^. because z c X^ then Z = ^nO^f^ e z and Gra, = ctq = a^a^ in X^. We calculate XOr = (EOE'')0(EOE') = (EOE)0{E'^ OE') = iEOE)O(EO'O'E') = {EOE)0{EOoE)' = (EOE)0(OEOOE)' = (EOE)00' = {EOE)00 = EOE e /C. YOT = iEzOEl)0{EzOEi) = {EzOEz)0(,EO OEz)' = E^OEz e /C,g. ZOr= (£^C'£--)0(£,iO£'^) = E^OEr^ e /C,n. Thus, if we put £, = OX, £2 = OY, £3 = OZ, and £4 = OT it follows that the set {Ei,£2,£2,£4} consists of four mutually orthogonal vectors such that £, + £4 is isotropic for / = 1,2,3. So if ^ is a symmetric bilinear form representing n then from ^, + £4 1 £, + £4 for / = 1 , 2, 3 and £/ 1 £j for / ?iy it follows that for / = 1,2,3, O = g{Ei + E4,£i + £4) = g{£i,£i) + 2g{£i,£4) + g{£4,E4) = g{£i,£i) + g{£4,£4). So that giEi,£i) = -g(£4,£4) ^ O, because £4 is nonisotropic. Thus, it remains to show that {£\,£2,£3,£4} is a basis for V. But this follows from Section 3.7. ■ Theorem 3.9.23. (V,}C,g) js a four-dimensional Minkowski vector space. Moreover, (.ty(0),g-) is a four-dimensional Minkowski space for every y in Vq- Proof: Put g(E4,E4) = -1 and g(£„£,) = 1, for / = 1,2,3. Minkowski space is the only nonsingular real four-dimensional vector space with metric g f 1 ^ 1 I lo -1 J (3.24) 86 In the last section of Chapter 3 we show that each A, in ^ can be identified with a spaceUke plane and each cx e Q* with a reflection about a spacelike plane. 3.10 Spacelike Planes and Their Relections For each X e g with Ol^i, define ax : V ^ V by axiOA) = (OA)^ = OA^, for OA e V. To extend this definition to any X, e ^, we note that if A. / O and OA e V, then there exists a unique D in X such that OO^A^ = D; that is, there is a unique D in X such that (OA)^ = O^A^ = OD. Thus, we define GxiOA) = OD, where OO^A^ = D. We note that if A | O then D = A^. First we show that 6x is a semilinear automorphism of V for each X e V. [23] Now, a function/: V i-> V is a semilinear automorphism if, and only if, it has the following properties: 1. /is an automorphism of the additive group of V onto itself 2. /sends one dimensional subspaces of V onto one dimensional subspaces (that is, / is a collineation). 3. If A and B are linearly independent vectors of V, the vectors XA) and/B) are also linearly independent. Theorem 3.10.1. The map ax Js an automorphism of the additive group ofV onto itself. Proof: Suppose that ax{OA) = ax(OB). Then ax(OA) = OD where OD = OM^ = O'-B^. This implies that ^^ = 5^ or ^ = B. Thus, OA = OB and ax is injective. To show that ax is onto, let OB e V and A = OO^B^. Then axiOA) = 0^A^ = 0\OO^B^)^ = O^O^OB = OB. We show ax is additive. Let OA,OB e V. Let D = OO^A^, F = OO^B^. Then axiOA + OB) = (OAOB)^ = O^A^O^B^ = ODOF = ax{OA) + ax(OB). ■ Lemma 3.10.2. The map ax sends one dimensional subspaces ofV onto one dimensional subspaces ofV. 87 Proof: Let < C14 > be a one dimensional subspace of V. Then there exists a line g = [a, p,y] such that OB e< OA > if, and only if, 0,B e g. Since 0,B I a,p,7 <^ 0^,5^ I a^,p^,y^ then d;,(< OA >) =< gx{OA) > is a one dimensional subspace of V. ■ Lemma 3.10.3. The transformation, di, maps linearly independent vectors ofV to linearly independent vectors. Proof: The vectors OA,OB e V are linearly independent if, and only if, goA '^ gOB if, and only if, go^A^ '^ So'^B^- Hence, &x is a semilinear automorphism of V for each x&g. m Theorem 3.10.4. The map ox : V ^^ V is a linear automorphism ofV onto V. Proof: Consider the definition of a semilinear automorphism [23,defn.73.11. Let (V,/C) and (V',/C') be vector spaces over the division rings K, and IC' , respectively. Suppose that ^i : /C H-+ /C' is an isomorphism from /C onto /C'. A map A : V i-> V' is called semilinear with respect to |x if 1. X{A + 5) = X{A) + X(B) for all A,B e V. 2. X(tA) = ^it)X(A) for all ^ e V and / e /C. The only isomorphism ^ : R >-► R is the identity. Thus, d;, is a linear automorphism of V onto V. ■ Let (V,/C,g) be a metric vector space. A similarity y of V is a linear automorphism of V for which there exists a nonzero r e )C such that g(yA,yB) - rg{A,B), for all ^,5 e V. If ^ is nonisotropic, r = ^^. The scalar r is called the square ratio of the similarity. Lemma 3.10.5. [23] A linear automorphism of a metric vector space V is a similarity if, and only if it preserves orthogonality. Proof: For all lines h and / in our space, h 1 I <r^ h^ 11^, hence, cxiOA) 1 diiOB) <^ OA ± OB. ■ 88 Theorem 3.10.6. The linear automorphism dx is an isometry. Proof: Let y : V i-^ V be a similarity with square ratio r ^ 0. Then for A,B e V we have g{yA,yE) = rgiA,B) = rg(y~\yA),y-\jB)) and giy-\yA),y-\yB)) = ]rg{yA,yB). That is, y"' has square ratio r"'. Now 6x is an involution, so that dx = ct^'. Hence, if A- 5^ is the square ratio of dx then r = -^ or r- = 1, so r = ±1. [23] Because the field K. is isomorphic to M, then V is not an Artinian space. Thus, for each similarity Ox of V there is an unique r > and there is an unique isometry a such that ux = M(P,r); where M{0,r)iA) = rA for all A in V. The map dx is thus a similarity with square ratio r^. Because r > 0, then from above, r = 1 and it follows that dx is an isometry. ■ Proposition 3.10.7. The isometry ax is a 180° rotation; that is, a reflection about a plane (a two-dimensional suhspace ofV). Proof: Let X & Q. Suppose that 0|?i and put O = Xa with a e M. Then for all A\X and for all B\a, dx(OA) = (OA)^ = O'^A^ = OA and Gx(OB) = (OB)^ = O^B^ = OB"^^ = OBO = -OB. Since ta(0) = tx{0)^ and V = .t^O) © tx{0)^ then d>. = l.t,(o) e -l.t,(o). Thus, ax is a reflection about the plane tx(,0). Suppose that O / A,. Let s | O such that s || X and let P | A. be arbitrary. Then if /? I s. FOR = Q \ X And OR = PQ. Thus, axiOR) = {ORY = (^0^ ^PQ = OR and ax V^^^o) = ^Uoy Now let PIO such that p 1 s, so O = sp. Let B \ p. Now pi?, because 8 || X and BOB = D\y where P = Xy and y || p. We calculate, ax(OB) = {OBY = {PDY = {PDy^ = {PDY ^ DP = BO ^ OB^ = -OB. Hence, ax = l.tj(O) © -1 ^^(^ji and ax is a reflection about a plane. ■ 89 To show that dx is a reflection about a spaceUke plane (EucUdean plane) for every X e ^, we use the following theorem from Snapper and Troyer [23]. Theorem 3.10.8. [23] Let {V,IC,g) be an n-dimensional metric vector space over a field K with metric g, K -'R, and n = 2. Every nonsingular real plane has a coordinate system such that the matrix of its metric is one of the following. ' 1 , f 1 , - the Euclidean plane, - the Lorentz plane, and \ ^ I -1 ' -10., - the negative Euclidean plane. -1 ' Hence, the Euclidean plane, the Lorentz plane, and the negative Euclidean plane are the only nonisometric, nonsingular real planes. This also follows from Sylvester's Theorem [23] (It states that there are precisely n+1 nonisometric, nonsingular spaces of dimension n). Lemma 3.10.9. Let /C = R, n = 4, andV be Minkowski space. Then the orthogonal complement of a Lorentz plane is a Euclidean plane. Proof: Let {e,},=i 4 be a basis for V such that the metric of V with respect to this basis has matrix of the form (3.24). Let m = 63 + e4 and m = 63 - 64 so that < m >,< n > are the unique isotropic lines in the plane < 63,64 >. Let a be a Lorentz plane with isotropic basis mi and np Then there is an isometry cj :< 63,64 >•-► a such that a(m) = mj and (j(n) = n,. By the Witt Theorem ]23] a can be extended to an isometry of V, which we also denote by a. This implies that a : V =< 63,64 > e < 61,62 »-^ V = a ©< a(ei),a(62) >; that is, {CT(6i),a(e2)} is a basis for a-*-. Now cj is an isometry and ^(ei,ei) = ^62,62) = 1, so the metric g with respect to a-^ has matrix I2, the 2x2 identity matrix, and a-^ is a Euclidean plane. ■ 90 Theorem 3.10.10. The map a^ is a reflection about a spacelike (Euclidean) plane for every X e Q. Proof: From the proof of Proposition 3.10.7 it suffices to consider X e Q with 0\X. Let O = QX with 9 e A^ and let Q{') denote the quadratic form associated to the metric g obtained in Section 3.9. Because Q e M there are precisely two isotropic lines ACe,, ICq-, c Xq. Let O ^ A e ICq^ and O ^ B e ICq^. Then OA and OB are isotropic vectors which form a basis for ^e(O) and Q(OA) = = QiOB). Therefore the metric of ^9(0) with respect to OA and OB has matrix g = f Oa\, . f r (OA OB) = y OB j^ ^ ^ r where the products of the matrix elements are the inner products defined by the metric g and g{OA,OB) = r ^ 0. Since -L • O^ e /Ce , , and OB e ICq^ then /Ce,, ICq^ also form an isotropic basis for .te(0). Hence we may assume that r = \. Let OTi = -j^iOA - OB) and OZ, = -^{OA + OB). Then g'iOXx.OT,) = 0, Q'{OXx) = 1, and Q'{OT^) = -1. Because OX, 1 OTx and Ae(0) is nonsingular then 0X\ and OTx are linearly independent and hence, form a basis for .te(0). Moreover, the metric of .te(0) with respect to OJiTiand OF, has the '^''^^''' 1 -1 ]■ '^^^'^^^°''^' ^y Theorem 3.10.8, .te(0) is a Lorentz plane and by Lemma 3.10.9, .tx(0) = te(0)^ is a Euclidean plane. ■ CHAPTER 4 AN EXAMPLE OF THE THREE-DIMENSIONAL MODEL This chapter begins by considering a net of von Neumann algebras, {7l{0)}oei, and a state co, coming from a finite component Wightman quantum field theory in three-dimensional Minkowski space. There are various senses to the phrase "coming from a Wightman quantum field theory". The assumption here is the version given by Bisognano and Wichmann [5]. That is, given a finite component Wightman quantum field, ^(x), assume that the quantum field operator, (j)(/), is essentially self-adjoint and its closure is affiliated with the algebra Tl(0) (in the sense of von Neumann algebras) for every test function /whose support lies in the spacetime region O. Driessler, Summers, and Wichmann show these conditions can be weakened [15]. But free boson field theories satisfy these conditions in three-dimensional Minkowski space [5]. For such theories the modular involutions, Jo, associated by Tomita-Takesaki theory to the vacuum state and local algebras of wedgelike regions, O, in three- dimensional Minkowski space, act like reflections about the spacehke edge of the wedge [5]. Since the modular involutions have that action upon the net, the hypotheses of Buchholz, Dreyer, Florig, and Summers (BDFS) are satisfied [6]. Therefore the Condition of Geometric Modular Action, CGMA, obtains for the set of wedgelike regions [27] in Minkowski space. The precise wording of this version of the CGMA is given below. Let /| and Ij be two lightlike linearly independent vectors belonging to the forward light cone in three-dimensional Minkowski space. The wedges are defined as the subsets WmJi] = {a/i + p/2 + /^ e R'-^ : a > 0, p < 0, (/^,/,) = 0, / = 1,2}, where ( , ) denotes the Minkowski inner product. 91 92 Let 1 1 =(1,1,0) and 1 2 = (1,-1,0) be lightlike vectors in R''^, three-dimensional Minkowski space, and let V be the Poincare group, the isometry group of this space. Then the set of wedges, W, is given by W = {A.ff[/i,/2] : A. e V}, where XW[li,l2] = {Hx) : X 6 W[lil2]}. The CGMA for Minkowski space is defined as follows. Let {TZ(W)} fy^w be a net of von Neumann algebras acting on a Hilbert space 7i with common cyclic and separating vector Q e H, satisfying the abstract version of the CGMA and where the index set / is chosen to be the collection of wedgelike regions W in IR''^ defined as above. Recall from Chapter 1, with ({TZiW}} h'^\\,\'H,Q.) there is the following. 1. A collection of modular involutions {Jw}iVeW- 2. The group JT" generated by {JH'}ivew- 3. A collection of involutory transformations on W, {'iw}weW- 4. The group Tgenerated by {Tw}iVew- Assume also that: 5. The group Tacts transitively upon the set W, that is, for every Wi,W2 e W there isaWi e yV such that iiv,ifV\) = W2. Note that this assumption is implied by the algebraic condition that the set {adJw}weW acts transitively upon the net {7^(^}jre>v- At this point the following two assumptions are added ([27] which have been verified for general Wightman fields). ■.•:•". 4.6 For W\,W2 e W, if W\ nWj i/^ 0, then Q is cyclic and separating for 'JZ(Wi)nn(W2). 4.7 For W^, W2 e W, if Q is cyclic and separating for n{W\)r\ n{W2), then W\nW2* 0. The CGMA for Minkowski space is the abstract version of the CGMA with the choice of W for the index set /, together with assumptions 4.6 and 4.7 above less the transitivity assumption [6]. 93 Buchholz, Dreyer, Florig, and Summers [6] showed that with the above assumptions one can construct a subgroup ^ of the Poincare group P, which is isomorphic to Tand related to the group Tin the following way. For each t e Tthere exists an element g^ e ^ such that x{W) = g-^W = {gx(x) : x e ^}. To each of the defining involutions Tjy e 7T ^ e W, there is a unique corresponding g^y e ^c V [6]. Moreover, BDFS obtained the following (suitably modified for three dimensions and abbreviated for our purposes). Theorem 4.1 [6] Let the group T act transitively upon the set W of wedges in R''^, and let ^ be the corresponding subgroup ofV. Moreover, let gw be the corresponding involutive element ofV corresponding to the involution xiv e T. Then gw is a reflection about the spacelike orthogonal line which forms the edge of the wedge W. In particular, one has gwW = W, ' the causal complement ofW, for every W e W. In addition, ^ exactly equals the proper Poincare group V+. M Recall from Chapter 2 that the initial model of (G, 0) is as a group plane. This means that each g e Q is viewed as a line in a plane and each P - gh, g\h, as a point in a plane. Let us call the axiom system given in Chapter 2 as - A. Thus as- A is & set of axioms about "points" and "lines" in a "plane". Let P denote the collection of points P e 0. For each a e define the map aa : P u ^ ^P u ^ by ^a{P) = ?"" = aPa-' for P e P and CTafe) = g"" = aga"' for g e g. Since {Q, 0) is an invariant system then each CTa is a bijective mapping of the set of points and the set of lines, each onto itself, which preserves the incidence and orthogonality relations, defined by "I", of the plane. We say that CTq is a motion of the group plane. Since Q generates then the set of line reflections g^ = {og : g e Q} generates the group of motions 0a = {cTa : a e 0}. Let O : -> 0<j be the map defined by 0(a) = aa, for a e 0. Then O is in fact a group isomorphism [2]. This 94 means that i'Q,<S>) is isomorphic to iGG,^G) (in the sense that Q is equivalent to Qa as sets and ^(Q) = Ga-, O(0) = ©a, where O is a group isomorphism). This implies that <^(as - A), which we denote by as - <t)(^), is an axiom system concerning the group of motions; line reflections of a plane, the group it generates, and point reflections of a plane. A plane whose points and lines satisfy as - A. As was shown in Chapter 2, given (^,(5) satisfying as - A, one obtains R*'-^, three-dimensional Minkowski space. Under the identifications given in Chapter 2, we find that each g e Q corresponds to a spacelike line in M''-^. Thus, as - <^(A) is a set of true statements concerning reflections about spacelike lines and the motions such reflections generate in three-dimensional Minkowski space. Moreover, since such motions are in fact isometries in R'--^ then 0(0) is isomorphic to a subgroup of the three-dimensional Poincare group. Theorem 4.2 Under the same conditions as in Theorem 4.1 it follows that {{\w}wey\>,'T) acting on W satisfies as- <i>(A). Proof: From Theorem 4.1 we have {{gw} w^w,"^) satisfies as - Q>i^A) since *p is the subgroup of V generated by reflections about spacelike lines. Also from Theorem 4.1, ({tjf} WeW,T) is isomorphic to {{gw} WeW,^) so {{\w} w^y^,T) satisfies as - O(^). ■ The net continuity condition assumed by BDFS [6] for the next theorem was later shown to be superfluous [8] for this theorem and the remaining theorems. Theorem 4.3 [6] Assume the CGMA with the spacetimeR^-^ andW the described set of wedges. If J acts transitively upon the set {7^(fF)} ff'eW then there exists a strongly fanti-J continuous unitary representation U(P +) of the proper Poincare group which acts geometrically correctly upon the net {Tl{W)}weW and which satisfies U(gn) = Jw, for every W e W. Moreover, U{Vl) equals the subgroup of J consisting of all products of even numbers ofJw 's and J = U{Vl) u Jw^ U{Vl), where Wr^{x & M'-2 :xx > \xo\}. m 95 Theorem 4.4 Under the same hypotheses as Theorem 4.3, the group J is isomorphic toV + = ^, which is generated by the set of involutions {gw \ W e W}. Moreover, {{Jw }w&v^,J) satisfies as - O(^). Proof: By Proposition 1.1 there is surjective homomorphism "t^: J ~^T, where the kernel of ^, ker(Q, is contained in the center of J, Z{J). By Theorem 4.3 there is a faithful representation C/(P+) such that f/(gw) = Jw, for every W e W. Since the center of V+ is trivial. t/(') is a faithful representation of Vi and hence an injective map preserving the algebraic relations, Z{J) is trivial. This implies that ker(^)= {1} and ^ is an isomorphism. If T : T-> *p denotes the isomorphism of T and ^ given by BDFS from Theorem 4.1 then T o £, : J" ^ <p = 7?+ is an isomorphism. It therefore follows that the pair {{Jw] WeW,J) satisfies as - 0(A). ■ We can now give the main result of this chapter. Theorem 4.6 Any state and net of von Neumann algebras, coming from a (finite component) Wightman quantum field in three-dimensional Minkowski space, which satisfies the Wightman axioms, provides a set of modular involutions satisfying as-^{A). ■ As a final remark we note that since free boson field theories satisfy the Wightman axioms and therefore the CGMA for Minkowski space holds, then these theories give a concrete example of the three-dimensional case of this dissertation. CHAPTER 5 CONCLUSION At this point it is useful to briefly recall the starting point of this thesis, to restate the problem, and to summarize the results obtained. It is assumed that there is a net of C*-algebras {Ajjiei, each of which is a subalgebra of a C*-algebra A, and a state CO on A. If this net and state satisfy the Condition of Geometric Modular Action, COMA, is it possible to determine the spacetime symmetries (the isometry group), the dimension of the spacetime, and even the spacetime itself, without any assumption about the dimension or the topology of the underlying spacetime? Recall also that the resolution of these questions involved two steps. First, given a set of involution elements G and the group © it generates, find necessary conditions on the pair (G, ©) that will allow a construction of three- and four-dimensional Minkowski space. Moreover, this should be done in such a way that the generating involutions can be identified with spacelike lines or spacelike planes and their respective reflections. This was done for three-dimensional Minkowski space in Chapter 2 and for four-dimensional Minkowski space in Chapter 3. The second step of this process is to determine what additional structure on the index set /would yield algebraic relations among the modular involutions, {J/},g/, such that the pair ({J/}/6/, J) satisfies the axiom systems given in Chapters 2 and 3. Using the work of Buchholz, Dreyer, Florig, and Summers [6] and the work of Wiesbrock [29], we are able to obtain a result concerning the above step in three dimensions. First we briefly recall the abstract version of the Condition of Geometric Modular Action, CGMA, described in Chapter 1. We assume there is a net, {7^,•},g/, 96 of von Neumann algebras acting on a Hilbert space H, where the index set I is a partially-ordered set. There is a vector Q e H which is cyclic and separating for each TZi, i e /. From the modular theory of Tomita-Takesaki we then obtain a collection, {J,}, -6/, of modular involutions which generates a group J" and a collection, {A,}/g/, of modular operators. The assignment i ^ TZj is an order-preserving bijection and each ad Jj leaves the set {7^,}/e/ invariant. The last two assumptions imply that for each /■ € /, there is an order-preserving bijection t, on / such that J,7?.yJ, = 7^t,(/> J e /. The group generated by {t,}/6/ is denoted by Tand forms a subgroup of the transformations on the index set /. Assume also that the two intersection conditions for wedges given in Chapter 4 also hold as part of the CGMA for what follows. To help explain the additional assumptions used in our result we give the following definitions and theorems. Definition 5.1 [6] The Modular Stability Condition (MSC). Let {'ll{W)]w^w be a net of von Neumann algebras satisfying the CGMA where the index set / is the set of wedgelike regions W in E'-^ described in Chapter 4. Then the modular stability condition is satisfied if the modular unitaries are contained in the group J" generated by the modular involutions; that is, A'^ 6 J for alU e M and W e W. Theorem 5.2 [6] Assume the CGMA for three-dimensional Minkowski space with (4.6) and (4.7), where the index set / = W, the collection of wedgelike regions in i?''2. Assume also the transitivity of the adjoint action of J on the net {n(W)} ^.^vv- Let f/(R'-2) be the representation of the translation group. IfA'^ e J, for all t e M and some W e W, that is, if the modular stability condition obtains, then sp(U) c V+ or sp(JJ) d V-. Moreover, for every future-directed lightlike vector H such that W+lczW, there holds the relation A'^U(t)A-^' = f/(e-«'f), for all t e R, where a = ±271. 97 Definition 5.3 [29] Let J\f, M be von Neumann algebras acting on a Hilbert Space Ti-Let Q be a common cyclic and separating vector in H. If A^' ATA^ c A/", for all / > 0, we call (M d M,Q) a +-half-sided modular inclusion (+-hsm). If Ajt( A/" A'}!^ d jV, for all r < 0, we call (Af c: M,Q.) a —half-sided modular inclusion (— hsm). Definition 5.4 [29] Let J\f, M be von Neumann algebras acting on a Hilbert space TC with Q e H a common cyclic and separating vector for A/", M., and A/'n Ai. 1. If ((AAn M) e A/',Q) and ((A/'n A^) c A/(,Q) are ±-hsm inclusions. 2. And if J^is - lim,^+oo A'ifA~/\)JAf = s- lim,-^+oo A'j^Aji'. Then we say that such a pair ((M, M), Q) has (±) modular intersection, ± mis. Theorem 5.5 [29] Let J\f, M, C, and U be von Neumann algebras acting on a Hilbert space H with a common cyclic and separating vector Q e Ti. Assume the following. I. l.(M,M,n) is- mis, 2. (£,M,D) is+ mis, 3. (£,A/',Q) is- mis, II. l.iMdMQ)is-hsm, 2. AdJM(Jj{fJM) = JMJj{f, 3. [AdJc{JuJM),Ju-^Mh 0' III. l.Ad (AdJciJj^JNY) {Ad A^V/;(JaA/^))(J^J^)2(A0 c M, with t a = ^ln2. Then the modular groups A'ir, A'jii, A'l, and A'j^, for t, r, s, v eR, generate a unitary representation of the 2+1-dimensional Poincare group. ■ Remarks [29] The conditions in I. give a unitary representation of the 2+1-dimensional homogenous Lorentz group. The hsm inclusion of condition II. equips us with a representation of the translations along some light ray. The product of the two modular conjugations is then a finite translation of this kind. Moreover, due to 98 the result of Bisognano and Wichmann [5], the modular conjugations of the wedge algebras act as reflections. These properties are encoded in condition II. 2 and II. 3. A physical framework will now be given as a description of quantum field theories in terms of local nets of algebras [17]. The basic assumptions are the following. Let {A(0)}oeV c B(H) be a net of von Neumann algebras indexed by the closed double cones V in M*'^ which satisfy the following properties: 1. (Isotony) If O, c Oj, then A(O0 c AiOi). 2. (Locality) If Oi c Oj, then AiO\) c AiOi)' . Where AiO)' denotes the commutant of A(0) in B(H) and O' is the causal complement of O c R''^. 3. (Poincare covariance) There is a unitary representation U : SO (2, 1) > R'-^ -> U(H) of the Poincare group with positive energy. 4. (Vacuum vector) There is a unique W -invariant vector Q e H. The algebra AiO), the inductive limit of the net, is called the local algebra of observables localized in O c R''-^. As was mentioned in Chapter 4, if the local net is generated by Wightman fields then the modular groups associated with algebras of observables localized in wedges act as Lorentz boosts in the directions of different wedges and the modular conjugations act as reflections [5]. In particular, the adjoint action of the modular conjugations on the net act as reflections about the spacelike edge of the wedge. For what follows we shall call the properties in the previous paragraph the Bisognano and Wichmann property. Theorem 5.6 [29] Let AiO), O c R''2, be a local net fulfilling the Bisognano and Wichmann property for wedges. Let M = A^/1,/2]), M. = A^/1,/3]), £ = AiW\li,h'\), andk^ AiWlU^h,-^-^, where hJj, and 1 3 are three linearly independent light rays. Then this set of algebras together with the vacuum vector Q fulfill the assumptions of Theorem 5.4. Conversely, let M M, £., and M be a set of A 99 VOD Neumann algebras acting on a Hilbert space H together with a common cyclic and separating vector Q. e H, which fulfill conditions I.-III. of Theorem h A. Then these data determine a local (Bisognano-Wichmann) net A{0) e B{H), O c R'''^, such that the incident algebras become the wedge algebras of the constructed net as in the first part. ■ Theorem 5.7 Let {Tli} j^i be a net of von Neumann algebras acting on a Hilbert space ^, together with a common cyclic and separating vector Q e H satisfying the CGMA with the MSC. Assume also that there exist iM,JM,kj{f,lc e / such that '^w^'^jM^'^kj,', andUi^ satisfy the hypotheses of Theorem 5.5. Then the following are true. 1. There is an injective map F : W ^I, such that for each W e W, the modular objects J F(W), ^'f(W) have the Bisognano-Wichmann Property when acting upon 'R.F(W)- 2. The modular unitaries A}'^,, A/^, AJ^^^, and A)^ generate a continuous unitary representation of 7^+ which acts covariantly upon {TIf(W)}w^w- If the additional intersection assumptions, (4.6) and (4.7), are made on the subnet {TZfifV)} iVeW, then: 3. The CGMA as stated for Minkowski space holds for the subnet {TIf(W)} w^w, as does the MSC. 4. The modular conjugations { adlpov) } w^m and thus {{xf(W) } We w, T), satisfy the axioms of Chapter 2. 5. There is a continuous (anti-) unitary representation of P^ acting covariantly upon Proof: For the proof of Teorem 5.7.1, we give the construction Wiesbrock gave in the proof of Theorem 5.6 [29]. Let /, = (1,1,0), h = (1,-1,0), /j = (1,0,1) e R''2. The local algebra of observables to wedges is defined by For arbitrary linearly independent light rays /,,/,■ e M'-2 pointing to the future, let 100 Al^j € iSO' (1,2) with // = A/^./ 1\, and Ij = Aj.j.lj. This element in S0^(\,2) is uniquely defined up to a multiplication by a boost of type Ai^j^{t), t e M with the given asymptotics l\,l2- Let U denote the unitary representation of the Poincare group according to Theorem 5.4, that is, let ZY(A/„/,(0) -Af;, U(Ai,i,it)) ^ A'ijl, U{A,^j^(t)) ^ A^iYc ' ^ ^- Now define the observable algebra associated with arbitrary wedges by n{W{liJj-\) = adU{Ai^j){Jlij^) ( e {7^,}/e/ by the MSC and the CGMA). For translated wedges, define for a e R''-^ n{W[liJj,a\) = adU{a)U{A,^j){nij^) ( e {Te;}/^/ by the MSC and CGMA). In this way, for any wedge region PF in R''^ there is a unique von Neumann algebra Uj^ in {7^,},•e/. Taking F : W ^ I to be the map F{W) = />,, iox W e W and /> e / as obtained above and the result follows. Conclusion 2 follows from Theorem 5.6 and Theorem 5.5. Conclusions 3 and 4 also follow from Theorem 5.6. The last conclusion follows from Theorem 4.3. ■ We conclude this chapter with a few remarks. Given a net {^,},e/ and a state to satisfying purely algebraic conditions, one derives three-dimensional Minkowski space, and an identification between elements of / and the wedges in three-dimensional Minkowski space in such a way that the adJi act like a reflection through the spacelike edge of the wedge. Therefore, solely with assumptions on the algebras of observables {^,},g/ and the preparation co, we are able to derive the physical spacetime and its symmetries. We can also derive an interpretation of suitable elements of {^,}/e/ as local algebras associated with wedge regions, as well as derive a prescription of how the spacetime symmetries act upon the observables. In addition, we can get a time orientation of the spacetime from the MSC. 101 A similar process can be done for four-dimensional Minkowski space using the work of Wiesbrock and Kahler [18]. However, we refrain from giving the details here. -'■ -t '-'■': r.- 102 REFERENCES |1] F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd ed., Springer, Berlin, New York, N.Y., 1973. [2] F. Bachmann, A. Bauer, W. Pejas, and H. Wolff, Absolute Geometry, Fundamentals of Mathematics, Vol. 2, MIT Press, Cambridge, Mass., London, 1986, pp. 129-174. [3] R. Baer, Linear algebra and projective geometry, Academic Press, New York, N.Y., 1952. [4] A. Bauer and R. Lingenberg, Affme and projective planes. Fundamentals of Mathematics, Vol. 2, MIT Press, Cambridge, Mass., London, 1986, pp.64-111. [5] J. Bisognano and E. Wichmann, On the duality condition for a hermitian scalar field, J. Math. Phys., Vol.16, 1975, pp.985-1007. [6] D. Buchholz, O. Dreyer, M.Florig,, and S. J. 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Lingenberg, Metric Planes and Metric Vector Spaces, John Wiley and Sons, New York, N.Y., Chichester, Brisbane, Toronto, 1979. [21| H. Noack and H. Wolff, Zorn 's Lemma and the High Chain Principle, Fundamentals of Mathematics, Vol. 1, MIT Press, Cambridge, Mass., London, 1986, pp.527-529. [22] A. Seidenberg, Lectures in projective geometry. Van Nostrand, Princeton, N.J., 1962. [23| E. Snapper and R. Troyer, Metric Affme Geometry, Academic Press, New York, N.Y., London, 1971. [24] R. F. Streater and A.S. Wightman, PCT, Spin and Statistics, and All That, Benjamin, Reading, Mass., 1964. [25] S. J. Summers, Geometric modular action and transformation groups, Ann. Inst. Henri Poincare, Vol. 64, 1996, pp.409-432. [26| M. Takesaki, Tomita 's Theory of Modular Hilbert Algebras and Lts Applications, Lecture Notes in Mathematics, Vol. 128 Springer, 1970. [27] L. Thomas and E. Wichmann, Standard forms of local nets in quantum field theory, J. Math. Phys., Vol. 39, 1998, pp.2643-2681. [28] R. White, A Group- Theoretic Construction Of Minkowski Z-Space Out Of The Plane At Infinity, preprint. [29] H.-W. Wiesbrock, Modular Intersections of von-Neumann-Algebras in Quantum Field Theory, Comm. Math. Phys., Vol. 193, 1998, pp.269-285. 105 [30] H. Wolff., Minkowskische und absolute Geometrie I, Math. Annalen, Vol. 171, 1967, pp. 144-164. [31] J.W. Young, Projective geometry, MAA, Chicago, 111., 1930. BIOGRAPHICAL SKETCH Richard K. White was born in Richmond, Virginia on August 19, 1960. He graduated summa cum laude from the University of North Florida in 1991 with a Bachelor of Science degree in Mathematics. He graduated from the University of Florida in 1994 with a Master of Science degree in Mathematics. In May 2001 he graduated with a Ph.D. in Mathematics from the University of Florida. He is the proud parent of a six-year-old angel, Jackie. > * . ' i**^ i 106 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Stephen J. Summers. Chairman Professor of Mathematics 1 certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gerard Emch Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. /^ . f-yC^'-^-JU^ JohA R. Klauder Joint Professor of Mathematics and Physics I certify that 1 have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Itc^ /h^-,. David Groisser Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and qualitv. as a dissertation for the degree of Doctor of Philosophy. OL^\LA^h Khandkera Muttalib Professor of Physics This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 2001 Dean, Graduate School r> Uj^bSl- UNIVERSITY OF FLORIDA 3 1262 08555 1967 '•'Jf-..